# Tài liệu bồi dưỡng học sinh giỏi chuyên môn toán

2,9
Giai. Xet/„ = j

H a i day tiep c a n n h a u

(sinx)"dx,tac6:
2 9.1

Mot

s6 kien thufc thifdng dung.

plnh n g h i a 2 0 . Hai day so ( x „ ) vd (2/„) gQi la tiep c^n nhau niu
/„ = - y

Jcosxd(sinx)"~\

( s i n x ) " - ^ d(cosx) = - ( s i n x ) " " * .cosx|| +

0

0

/„ = ( n - l ) y " c o s 2 x ( s i n x ) " - 2 d x = ( n - l ) J
0

/„ = ( n -

{1 - sin^ x) ( s i n x ) " - ^ dx,

D i n h l y 18. Hai day tiep cq,n nhau thi hoi tu den cung mpt gidi han.

0

f

f

1) y

(sinx)"-=^dx - ( n -

1) J(smx)''dx

0

1)4.

= (n - l)/„_2 - (n -

Cbufng m i n h . Gia sii ( x „ ) , (j/„) la hai day so t i l p
nhau
( x „ ) la day
tang, (yn) la day so giam. Ta cut cho i&ng ba dieu kien (1), (2), (3) dung ngay
tif n = 1 (neu khong t h i t a load di mot so so hang). Xet day so (zn) dinh bcti

0

= x „ - j / „ , V n = 1,2,...

V?Ly n/„ = (n - l ) / „ _ 2

/„ = ^ ^ / „ _ 2 . V i /Q = | , / i = 1 nen:
2n - 1
2n - 1 2n - 3 ,
2n
• 2- ; r1- 22n
n -- 23- " - 43- 21- TT
2'
2TI ' 2 n - 2 2n
2n-2
4 2^
"'2n
^
_2n_
2n-2
2n + l ' 2 n - l " 5 ' 3 '
2n
+
r
2
n
-l
2n + l

Ta CO
Zn+i - 2n

= (x„+i - J/n+i) " (x„ - y„) = (x„+i - x „ ) + (y„ - j/„+i),Vn e N * .

Ma x„+i - x„ > 0, Vn = 1,2,... va 2/„ - y„+i > 0, Vn = 1,2,... nen
Zn+i

Do (16:

'

2n

/2n-l ~

2n+l'

ViO
0;|

V\$y (z„) la day tang va hpi t u ve 0. Do do Zk <

J

r/2

(sinx)2"+idx< y

0

Ma

lim

2n

(sinx)2"dx< J

< hn-l

=•

{sinxf"-'^

nii!?oo 1^"

dx

0

2n + 1

=

T

340

= 0 suy

ra

l i m (x„ - yn) = 0 4^

2hnn-l

< —

^2n

= ln5nta(-)tac6^1im^^ =

2.4.6...2n
3.5.7...(2n-l)•

~

n—>+oo

1

V2Hn

^

< l-

l.TCr(*)suyra

n-^+oo 2n + 1

JSfoo

(x„) tang va bi chan tren (bi c h ^ tren bcii yi chang h?in) nen hpi t u .
(yn) giam vk h\n dudi (bj ch?in diTdi bdi x i chang h^n) nen hpi t y .

n/2

0

=*-/2n+l < hn

n—+oo

x i < X2 < X3 < • • • < x„ < y„ < j/„_i < • • • < y2 < y i %

0<

n—•-j-oo

l i m 2 „ , VA; = 1,2,... hay

^fc<0,Vfc = l , 2 , . . . , d o d 6 x f c < j / * ; , V i b = l , 2 , . . . N h u v?iy t a c6

nen ( s i n x ) ^ " ^ ' < ( s i n i ) ^ " < ( s i n x ) ^ " - * . Do do

»/2

l i m \zn\ 0 hay lim z„ = 0.
n—»+oo

[3.5...(2n-1)1^2x1+1)

/2n+i

-z„>0,Vn=l,2,...

Do do ( 2 „ ) la day so tang. T i t (3) suy ra

(*)
"7^"

chung

cldng thdi thoa man ba dieu kien sau:
(1) (xn) la day so tdng.
(2) (2/„) Id day so gidm.
(3) l i m | x „ - 2 / „ | = 0 .

fl

(

'

^han xet 22.

l i m x„ =

n—»+oo

l i m j/„.

n—»+oo

Neu hai day so (x„) vd {y„) dong thdi thoa man ba dieu hen

(1) ( x „ ) Id day so tdng,
p ) (j/n) Id day so gidm,
(3)x„•^aj day ( x „ ) ud (y„) /ipi tv vd

l i m x„ <
n—>+oo

341

lim
n->+oo

yn-

2.9.2

C a c bai toan.

,

2
_ _]_ ^ 2 - (n + 1) _
1- n
(n+1)!
n!
(n+1)!
" ( n + 1 ) ! ^ 0 ' ^ " = L 2 , • •.

^,

B a i t o a n 234. Cho hai day so { a „ ) va (6„) nhu sau:

V\$.y day ( i ' n ) la day giam. Ta cd
0 < 6 , < a , ; a „ + i = ^^4^,

6„+, =

(Vn = 1,2,...).
2

dn + On

lim

Chiing minh rang hai day so da, cho hqi tu va gidi hg,n cua hai day so do bduy
nhau.
G i a i . R6 rang a„ > 0, 6„ > 0 (Vn = 1,2,...). T i t 2 {al + bl) > (a„ + b„f

\un-Vn\=

n—>+oo

lim

^ = 0 .

n — ' + 0 0 n!

• i 'I

Po do (Ufi) v a (t;n) l a hai day ticp can nhau. Suy r a hai day nay h o i t u den
cung mPt so, t a k i hieu so do l a e. Ta cd

suy r a a„ > 6„, Vn = 1,2,... T a c6

" n < e < t^n, Vn = 1,2,...

On+i =

Ta gia suf phaii chiing rang e la so hOXi t i , e = — , vdi p v a no la hai so t u

-T- <
a„ + 0„

— r — = a„, Vn = 1,2,...
a„ + &„

nhien, no khac khdng. Ta cd

Suy r a day (a„) la day so giam. T a c6

1 1
1
p
,
1
1
l + 7r + ;77 + -- - + — r < — < l + 77 + r T
1!
2!
no!
no
l ! 2!

1
+

-- - +

— 7 +

no!

1
— : .

no!

Nhan t a t ca cho no! t a dudc
Suy r a day (6„) la day so tang. Vay
6i < 62 <

> l < p ( n o - l ) ! < A + 1,

•• < 6„ < a„ < ••• < a i .

Do do day so (a„) giain va bi chan dudi (bdi so 61), day so (&„) tang va bi
chan tren (bdi so o i ) . Vay ca hai day so da cho cung hpi t u . D a t
va

lim

a„=a

l i m b„ = b, k h i do tijf

n-»+oo

(*)

trong do A la so t u nhien, >l = no! + ^ + ^ H
1M a (*) khong the
1!
2!
no!
xay ra dUdc (vi A\k{A-\-1)
la hai so t u nhien lien tiep, p(no - 1)! cung la so
t u nhien). Dieu vo If nay cluing to e khong the la so hiiu t i . V\$,y e la so vd t i .
Bai toan 2 3 6 . Cho day so (a;„) nhii sau :

6„4.i = ^4^

(Vn = l , 2 , . . . )

cho n —• +00 t a dildc b = —^—, hay a = b. Vay t a cd dieu phai chiing minh.
B a i toan 235 (De nghi t h i O L Y M P I C 30/04/2003).
va (vn) xdc dinh bdi:

H i f d n g dan. Xet day so (?/„) n h u sau 2/„ = x„ + J ^ ' ^ ' ' ^ - = 1.2,. • • K h i dd

Wn = 1 + Tf + ;7f + •• • + ^ ,
1!
2!
n!

i;„ = u„ + - ^ , V n = 1,2,...
TV.

Chiing minh (u„) va (u„) cd gidi hQ,n chung la mqt so v6 ti.
G i a i . u„+i -Un

= -.—^—rr-.

(n + 1)!

Chiing minh r&ng day so nay cd gidi han la mQt so vd ti.

Cho hai day so (u„)

> 0, Vn = 1,2,... Suy r a day (ti„) la day tang-

^^cng t u n h u bai toan 235 t a chiing m i n h dir0c ( x „ ) va (y„) la hai day tiep
•^ta nhau va hai day so nay cd gidi h^n chung la so vd t i .
^ a i toan 237. Cho day so {x„}+f°i nhu sau:
x„ = V ^
+ - ^ , V n= l,2,...
^
k\!
k=o
Ch '
''^^9 minh r&ng day so nay cd gidi hfin la mgt so vd ti.

342

343

HUdng d i n . Xet day so

" h u sau:

{yn}n=i

"

V^y day (xn) tang va bi chan tren (bdi yi chang h^n) nen hOi tu, day (y„)
giain va bi chan dudi (bdi x i chang han) nen hgi ty. Ta c6

1

lim

fc=0

y„ =

lim

n—+00

Khi do titcmg tit nhit bai toan 235 ta chi'mg minh diroc (a;„) va {y„) la hai
day tiep c§n nhau va hai day so nay c6 gidi hgn chung la so v6 t i .
{yn}t^i

Bai toan 238. Xet hai day so {xn}n=i.

1+ -

n—+00 \

=

ny

lim ( l + - )
n—+00

\

71/

. lim ( l +
n-.+oo \

i)=

Um x„.

n-.+oo

nj

V^y ta CO difiu phai chi'mg minh.

nhu sau:

isjhan xet 23. Tic (*) ta thay
x„=(l + i)",

iM=(l + i ) " '

(V„=.,,2,...)

n+l

,Vn=l,2,...
ChUng minh r&ng:
a) Xn < 2/n,Vn = 1 , 2 , . . .
b) Day (x„) tdng thuc si/ vd ddy (2/„) gidm thtfc stf.
c) Hai ddy so {x„}+~i, {yn}t=i
cimg gidi han, duac goi Id so e.
Giai.

Bai toan 239. Cho hai ddy so (x„) vd (y„) nhu sau:

a) Hien nhien.
b) Sur dung bat ding thiic Cauchy cho n + 1 so, gom mpt so 1 va n so 1 + ta dildc:
1 +n
1 + -

Mgitdi ta chiing minh dUcfc rang e Id so vd ti vde = 2,7182818284... Logarit
c(j so e goi la logarit tu nhien vd ky hieu Inx = log^ x. So e dong mot vai trd
rat quan trQng trong todn hgc. SvC dung {**) ta gidi dugc hai bdi todn kho {bai
todn 239 vd bdi todn 240).

<

n + l

a.,.a

n + l

x„ = l + i + ...+ - l ^ - l n n , y„ = l + i + ...+ _ l _ + i _ i n n ( n = l , 2 , . . . ) .
Chatng minh rdng hai ddy so da cho hoi tu.

1+ -

1 +

n+l

Hi/dng d i n . Tfir (**) 6 n h ^ xet 23 t a c6:

dang thiic Cauchy cho (n + l ) so, gom mgt so 1 va n so 1

ta dUdc

< l n e < (n + 1) In

n\n^^^^
n

Vay x„ < x„+i, Vn = 1,2,..., nghia 1^ day (x„) tang thvfc sy. L^i diing b a t

Hay vdi mgi n = 1,2,... ta c6: In ^

Vn = 1,2,...

n

< 1 va ^

< I n ^ . Do do vdi

'^Wn = i , 2 , . . . t a c o
^n+i

- x„ =

in

- ln(n + 1) + Inn =

2/n+i - y„ =

\

J

\ n - \

\

\

in -

In

n

- ln(n + 1) + Inn = —

n+\l

> 0,
In ^ ^ - ^ < 0.

'-

n

ta .se chiing m i n h diMc hai day so (x„) va (y„) la h a i d a y tiep c a n n h a u

Vay y„ < y„_i, Vn = 2,3,..., nghla la day {y„) giam thi^c s\f.
c) Theo tren ta c6:

1 Cling tien den ciing mot gidi han. Tit do ta c6 dieu p h a i chiing m i n h .

toan 240. Cho hai ddy s6 {x„} vd {y„} nhu sau:

X I < X 2 < • • • < x„-i < x„ < y„ < yn-i < • • • < y 2 < yi, Vn = 1,2,...

344

345

g l i i t o a n 2 4 1 . Xet day so { u „ } duac xdc dfnh bdi u„ = n^" (n g N*). DCit

w„ =
^
4^ =
+ • •• +
,
^
=,Vn = 1 , 2 , . . .
^"
^ n ( n + 1)
v / { n + l ) ( n + 2)
v^2n(2n+l)
l i m x„ ud

Hay tlm

n—"+00

lim

1 , 1

y„.

n—•+00

Chihig minh rdng day so

G i a i . TCr 16i giiii bai toan 239 t a c6:

t,a gi(ii hO'^ '^^

^ < l n ^ < i , V n = l , 2 , . . .
+ 1

n

Do do:
1
- +
n

1
- +
n n

1 .

n

{ x „ } + f ^ c6

gidi han hQu h(in khi n tdng len vo h^n

so vd ti.

G i a i . Ta c6

n

> 0 , V n = 1,2,...
1

1
,
+ •••+ — < In
2n
n
1
1
, n
+ •••+ — > In
+ 1
2n

n + 1

Vay

n
- 1
+ l
n

, n + 1
, ,
2n
,
+ In
+ • • • + In
=
n
2n-l
, n + 2
,2n + l , 2
+ In
+ • • • + In —
=
n+ 1
2n

2n
In
-,
n - 1
n + l
In
.

l „ ? ! ! d l l < x „ < l n - ^ , V n = 2,3,...
n
n - 1

Hm

(hi ^"

lim

^ ^ =ln(
^

/

HlLlLi^

\^n—'+00

n

=ln2;

/

<1 +

=ln2.

n — 1J

Vay tir (1) siif dung nguyen ly kepsuy ra l i m a:„ = l n 2 . V6i mpi n = 2 , 3 , . . . ,

I

I

1

1
23

/ I
22

2"

x / n ( n + 1)

1

v / ( n + l ) ( n + 2)

v^2n(2n+l)

1

1

1_

n n

+ 1

2n'

= 1 +

n2";

l - i

22

tu.

Ta chiing m i n h day { x „ } hpi tv bang each khac nhiT sau: Xet day so
nhu sau: y n = a;„ H

{yn}n=i

, Vn = 1,2,... K h i do

2/n+l -yn

1

,

{xn)n=\g va b j ch\$,n t r e n nen hpi

n—»+oo

ta c6:
1

,

— + — + ••• + — 1 = 1 +
V"2
"3
"n

V^ly day so
"^^

^ ^ ^ t khac v6i m p i n = 1 , 2 , . . . t a c6:
/

(1)

lim f i n
n—+00 \

{xn}t=\y

n

V i ham so / ( x ) = h i x Hen tuc tren khoang (0; +oo) nen

n—+00 Y

Y|ty

=

/
[Xn+l

N ,
X„) +

-

(n+1)

1
Un+l

1
Un

=

2

1

Un+\„

2n+»

m\$,t kliac

y ; i ( ; m ) ^ y ( n + l ) ( n + 2)
1

1

va

y2n(2n+l)

ra

xn+i

-

r - ^ r < Vn <
2fi +

l i m y„ = In 2.

lim — =

l i m 4 ^ = 0. Tom l ^ i (x„) la day tang,

l i m ( y „ - x „ ) = 0. Vay (x„) va (y„) la hai day so tiep

n—+ao

nli+

^" = ° '

^^"^ 1 < x „ < 2, Vn = 1 , 2 , . . . va day (x„) tang suy

2

C h u y 3 0 . Ta duac phep sii dung cong thilc sau: Cho
{xn}t=\
so nhdn c6 cong boi q thoa man dieu kien - I < q <1. Khi do ta cd cdc cdn9

"^a 1 < a < 2. Tiep theo t a chiing m i n h a la mpt so v6 t i . Gia sii ngupc Igi, a
so hQu t i : a = ^, vdi p, (/ G N*, (p, (?) = 1, p > g (do a > 1). Vdi moi so t\S
q
'''"^n m > 2 ta c6:

sau:

n
X1+X2

=

nhau nen ciing hoi t u den mot so, suy ra day (x„) hpi t y .

x„,Vn > 2. TiJ day sut dung nguyen l y kep suy

n—'+00

thiCc vd ky hi^u

-

la day giam,

1

>;r+T'^^^""^2n + l '
Do do

lim

+ - • •+x„+x„+i+- • • = ^

= J m ^ ( x i + X 2 + - • •+x„) =
346

- -

lim

V —=

lim

— + • •+

^Hm^l3^^'
347

] + 2^ —

Do do

WlU2..Wr-l

,

<

"^"^w

+i.Vj = 0,1,2,...

gdi vay nen vdi mpi j = 0 , 1 , 2 , . . . t a c6
"

+

+— ^ +

1

y —•

lim

'

{U\U2••-Ur-l)

Ihn V J - V

l i m y^^^-^-'-^r-x

Suy ra

"

hm

1

>

r i +^ .•••. +
.^_J_l>o.

p

<

Suy ra

U1U2

u

lim

y

n—+00 ^-^

k=m

}^hi A: nhan cac gia t r i 0 , 1 , 2 , . . . , n t a cho j = k, t a se c6:

- i . q l i m E — la mot so nguyen dudng, do do

J

n

(U1U2 • • - " r - l )

lim

(2)

n—•+(»

U1U2
fc=m

Hay
(1)

(2) suy ra ton t a i

= r > 2 di

fc=m

(urU2 . .

Tiep theo t a se chilng m i n h
V-

3m>2:

"i«2

• •-Wm-l-

lim

1

2^ - <

1

fc=m

d6 daii d i u mot diSu mau thuan. Goi r la s6 nguyen dUdng nho n h i t sao cho
r +j > 9 + l , V j =0,1,2,... Khidotaco
i2' < r 2 \ V i = l , 2 , . . . , r - l .

lim

1

< -.
7

Dieu nay mau thuan vdi (1). Vay a la so v6 t i .
L\iu y. Sau day la bai toan 242, t6ng quat cua bai toan 2 4 1 , day la bai toan
rat sau sic va kho nen doi hoi ban doc phai c6 nang luc va sU kien t r i dang
ke. Tuy nhien c6 mgt thuan Idi la trong bai toan nay c6 nhieu l?lp luan da
c6 6
Bai t o a n 2 4 2 . Cho {ukj^^i
J»n

= + 0 0 . Chy:ng

la day dan di^u tang
minh

rdng

cdc so nguyen

day so (a;„)+f°i, vdi

V i vay

U1U2

••-Ur-l

.2'+22+-+2'^ ^

"

.2'-2

(n=l,2,...)
fc=i

Ur+j

Ur+j

1

x„ =

vd gidi

han cua day (a;„)+^i la rnqt s6 v6 ti.

'^•ai. T a.,CO
Suy ra

x„+i - x „ =

> 0 , V n = 1,2,...
"n+l

<
349

dMng

sao

m

l a day s6 tang. T a d n g t u nhit bai toan 2 4 1 , t a cln'tng n i i n h cUtgc

{x,y^i

day

c6 gidi h a n hCtu han.

= +00

Jiin

nen J i m ^

= 0,

f hVfc v\$.y neu u i < 02 t h i clipn vi - 2. Neu o i > a2 t h i do J i m a*: = + 0 0 nen
.^i fc du Idn t h i ak > a i , do do chon r i la so nguyen duong nho nhat Idn hdn
2 sao cho a i < a^,. Thi^c ra ton t ^ i day v6 h^n cac so nguyen dUdng

D o do ton tai fci, fca,..., fcn sao cho

CO tinh chat nay, tiJc la O j < a^^, Vj = 1,2,...,

- 1. De t i m

{Tk)k=\

t a d p dyng

each tren cho day so {an\X=n^ • Gpi r la so nguyen duOng nho nhat sao cho

1

ar+j > 9 + l , V j = : 0 , l , 2 , . . . v a a j < a^, Vj = 1,2,..., r - 1.
Dat
a >

lim

= a, k l i i do t f l x„ > 0,Vn = 1,2,... va day (x„) tang suy ra

TTIIP theo t a chimg minh a la mpt s6 v6 t i .

hQu t i : a = ^, vdi p,g 6 N%

Gia

ngUOc 1^, a la

.5

Vi {w*}fc:^ 1^

= 1- Vdi m6i s6 tU nhien m > 2 t a c6:

ar = ( U r ) ^ < { U r + j ) ^ = ( « ' + / ) ^

Q
lim
— =
I™
n-.+oo ^
Uk
n->+oo
fc=l
:

lim
n—+00

— + ••• +

yUl

Urn

,

.}_,...+-J—

lim >

n->+oo
Ufc
*:=Tn

"»n-l

+

lim

y

"r+j

1

/

"

lim y^-!-

\

1

=

V ^1^2••-^r-l

lim

k=0
lira

1

*=0| (g + 1 ) ^ + 1 u,+jt

KiU2...u„.-i.q^^lim^2->l

Khi it nh?Ln cac gia t r i 0 , 1 , 2 , . . . , n t a cho j = fc, t a se c6:

(1)

fc=ni

(«.U2...,.._0(^„hm,l.—y

fc=m

T i 6 p theo t a chumg minh 3m > 2 : u i u j .. " m - i - „hm^ 2^

fc=m

= ^V^,

= 0' ^' 2. • • •

"

"

dgn d i l u man t h u l n . Dat

{q + 1)2^+'

la mot s6 nguyen duong, do do

^

+ iP^'

Bdi v^y nen vdi uipi j = 0 , 1 , 2 , . . . t a c6

> 0.

("i«2...a.-,)
£

(9

U1U2 • • - U r - l

fc=m

71—•TJu k=m

a2;y

Do do

—.

E - = --

<

Ur+j

1

S u y ra

N h u v^y u,u, .. - n . - . . 7 „ l i m

V-2

r.2'-2

"

lim

= " r ^ ^ V j = 1 , 2 , . . . , r - 1.

Viv?ly t a c o :

V

=

'^'^'^ ''•'^^S faf^ •''o nguyen duong nen Ur < " r + j , do do

g'

aj < a r , , V j = 1 , 2 , . . . , n - 1 -

350

y

\-

"•^^ (2) suy r a t o n t ^ m = r > 2 d ^
^.

t i t gia t h i l t suy ra ^ lirn^ ak = + 0 0

se chi ra rang ton t?ii so nguyen dUdng r j sao cho

lim

/
(uiU2...u„_i)

"

lim y"—
\n-.^oo^^Uk

"ay mau thuan vdi (1). Vay a la s6 v6 t i .
351

1 \

(2)

B a i t o a n 2 4 3 . Cho day cdc so nguyen {an}t=i

tf^oa man diiu hen

1 < a„ < max { 1 , n - 1}, Vn € N * .

Tirn diiu kien can va du di so a =

^

lim

^ " - ;V

i

la mot so hvCu ti

lim •

ni

fc!

n — + 0 0 .it—'
A:=ni+1

-

+ (ni-l)!

= 1

-

+ (ni-l)!

ni

"

V

— + ( n i - 1)! l i m

1

— e Q. T a chiing m i n h rling k h i

n—+00

2

= 1

Giai.
D i g u k i e n ckn. Gia sH a =

m
±
' - ' ^
iliim
n->+oo fe=rn+l

r

ni!

lim

y

(

^

Kl

3no e N* sao cho a„ = n - l , V n > UQ. Gia s i i phan chiing r i n g vdi mpi
no € N * , t o n t a i n > no sao cho a„ /

n - 1. D o a € Q nen a =

—,

vol

= l

ton t a i ni > 2M sao cho a „ , / n i - 1, tiic la 1 < a„j < n i - 2 . Ta c6

-

1

Vav 0 < ( n i - 1)! l i m

•/
2

-+ (ni-l)!—= 1 - ni

fc=ni

f - ^ - An ! yl

lim

n-.+oo \^ni!

2

K, M eN*. Suy ra vdi moi n> M t h i n!a € N * . Theo gia thiet phan chiing,

k=l

rii

nj!

1

+

ni

ni

1

= 1

< 1.

ni

5^ T T < l i d i i u nay mau t h u i n vdi
n

(n, - 1)! l i m T

^

e

N\

fc=ni

Ta CO ( n i - l ) ! a 6 N* va vdi mpi n e ( 1 , 2 , . . . , n i - 1} t h i ( n i - 1)! chia het
cho n ! . Do do

Man thuan nhan ditpc rhi'mg t o rKng 3no 6 N* sao cho a„ = n - 1, Vn > no.
Dieu k i e n d u . Gia sii 3no e N* sao cho a„ = n - 1, Vn > no. K h i do
n
(no + l ) ! a = (no + 1)! l i m

fc=i

V

fc=ni

^

>i—+CX) ^

(ni - 1)!
/J
m-l

= (no + 1)! l i m

(ni-l)!a-(ni-l)!5;^^

kl

n

"0

,

, fc! ^

2^

fc!

• no

=

M a t khac do 1 < o„i < n i - 2 nen t a c6:

lim
T»-« + 00

k=ni

~ 1^

fc=ni+l
fc=ni+l

V

Ti

+ (no + l ) ! hm

= ( n i - 1)!

^ ^

M

I

m!

+

a„i + 1

ni

m!

hm

n-.+oo

1

> ,

K!

/
n

- +

m

hm

n->+oo

\

>

n-+oo,

fc=ni+l

352

_v^a.(no + l)!

TT

fe=ni+l

—-

fc=no+l

n
= ( n i - 1)!

>

T7
fc!

fc!
fc!

/

,

,

ni

nm

k=l
no

E

/

l i

fc=no+l

rn

+ ("0 +

Kl

^

^
>

fc^

+
353

1)! h m

—r

- -r

n-*+oo

\no'.

n!

.—

^ ^

ajAno + ! ) ' • ^

Z-^

To

+ 1 ) 6 iV* C

c & i t i m A sao

cho

fc!

6 + Ad = A (a + Ac).

j
k=i

do ( y n + A 2 n ) ; t ^

thanh mot

cap s o n h a n , do do t a xac d i n h dudc

so h ^ n g t6ng q u a t c i i a day (?/„ + Xzn)'^^^,
Vay (ticu kicu can va d u (le a - ^ n m ^ ^

^, ^

q u a t c i i a h a i day

"

N h a n xet 24. TO kit qua bdi todn 243 suy ra e = J u n ^ E

1

yn+2 = ayn+1 + KcVn

^ ^ "Q-

p h i T d n g t r i n h dac

D a y s o phan tuyen tinh

{zn)'^^^

neu 116 ton tai.

CXn + d
Khi do day so (x„)+rj goi la day so phan tuyen tinh.

Q u i \idc. Neu clio (x„)^^j la day so phan tuyen tinh thi ta higu rang vdi
nipi n = 1 , 2 , . . . luon ton

ay„)

triTiig c i i a day

so {y„}

la

Giai p h i r d n g t r i n h dac; t n m g t r e n t a t i m d U d c so h a n g t o n g q u a t y„ ciia day
so {i/n}i v a tiir do t a t i m duoc so h a n g t Q n g q u a t 2„ ciia day so {zn}-

D i n h nghia day p h a n tuyen t i n h .

1 Vn € N*, x„+i = — — ^ ,

+ dz„) = a2/„+i + 6cy„ + d{y„+i -

A^ - (a + d)A + (ad - 6c) = 0.

Cho a, b, c, d thupc R sao cho ad - 6c 7^ 0 va c 7^ 0. Xet day so (x„) nhu sau:

2.10.2

day t a t i m dildc .so h a n g t o n g

= (a + d)2/„+i + (6c - od)y„.
Vay

2.10.1

(y„)+f°i va

tit

e a c h 2. Ta CO

a„ = n - l , V n > no-

2.10

so

(i)

x-„.

X a c d i n h so h a n g t S n g q u a t c u a d a y so p h a n t u y e n t i n h .

C h u y 31. Neu bd = 0 thi ta c6 the tuyen tinh hoa day so phdn tuyen tinh
(xn) dd cho, rSi sau do tim dU0c so hang tSng qudt cua day so (xn), lam cdch
my nhanh gon hdn. Ngodi ra ta cung c6 the tim dUdc so hang tong qudt cua
day so phdn tuyen tinh bang nhftn.g cdch khdc hay han nila, ban doc co thi
xem them bdi todn 30 d trang 2 4 {phudng phdp ham lap).
2.10.3

D j n h ly 19. Cho a, 6, c, d € M sao cho ad - 6c ^ 0, c ^ 0. Cho xi eR
mqi n sau:

Xet day so phan tuyen ti'nh { x „ } xac dinh bdi:

M o t so t m h c h a t c i i a d a y so p h a n t u y i n t i n h .

1,2,...,

ddt

va vdi

, = x „ + i , neu no ton tai. Xet ham so fix) nhU
cx„ + d

ax,i + 6
ox + 6
CX

trong do a, 6, c, d va p la cax. hang s6 cho tnfdc. Bang phudng phap quy na,
ta c L g nunh duoc x „ ^ £ , V n = 1 , 2 , . . t . o n g do

2/1 = p ,

thoa man

+ d

Chiing minh f Id song dnh.
I Cho day so ((„)

duoc dinh nghia bdi:

1 / yn+i = a y n + | ; 2 n , V n = l , 2 , . . .
zi = 1,| 2,^^, = c y „ + d2„,Vn = 1 , 2 , . . .

Vay van d^ con lai la t i n i s6 hang tfing quat cua hai day s6 (y„):ri
thoa man ( * ) . T a t h u d n g dung haicax^hnha sau:
e a c h 1 (ap d u n g k h i (i) c6 h a i nghiem p h a n b i e t ) . T a co
y„+i + A2„+i = (a + A c ) y n

+

354

(6 + Ad) 2„,Vn = 1 , 2 , . . .

tn+i = rHtn),yn

(^n)»=

=

i,2,...

'tdj/ CO the khong xdc dinh ki tit mgt thU tU ndo do. Con f'Htn)
SQ,U:

rVM

= | x e R \ | ^ } : / ( x ) =

355

(„}.

la tg.p

V^y vdi f la song anh thi dang tMc <„+i = f~^{tn)
nghla la tn+i la so duy
nhat di cho / ( t „ + i ) =
ChUng minh r&ng {xn)X^\ day phan tuyen tinh
khi va chi khi xi ^ t„, Vn = 1 , 2 , . . .
Chufng m i n h .
a) Theo gia thiet f{x)

/
-d\
\
la ham so xac dinh tren ( ^ - o c ; - j u ( ^ - ; + o o j v a

fix)

ax + b
-d
= — n - ^ ^ ^ "TCX + d
c

suy ra X3 khong xac dinh, niau thuan. Vay xi / <2- Neu xi = ^3 thi:

XI = rHt2) ^ f{Xl) = t2
=»a;2

^

/(X2).

xi

mau thuan ydi gia thiet. V?Ly X2
^ /(X2)

Ma ad - be ^ 0 nen suy ra xi = X 2 . Den day ta gJip man thuan. Vay

dy
cy-a

=
X2 =

^ / ( X i ) =t2=^Xi=

f{xi)

khac / ( x 2 ) . Do tto / la ddn anh. Tiep theo ta chi'mg minh / la toan anh. V6i
( 0.'\ " I " 6 r r
moi J/ e R\, xet phUdng trinh y =
T a co
b-

xi = f-^ih) =s> X i = <2
ti, suy ra X3 dUdc xac dinh. T a c6

X3

•v

axi + b _ ax2 + b
cxi + d
CX2 + d
<=>acxiX2 + adxi + bcx2 + bd= acxiX2 + adx^ + bcxi + bd

ih)
r\t2)

=^X2

=

t2

^Xi=h

mau thuan vdi gia thiet. Vay X3 ^ «i, suy ra X4 diidc xac dinh. Tudng
tu, ta CO Xj 7^ f 1 (i = 4 , 5 , . . . , n - 1) vi neu Xj = ti thi x, = U, mau
thuan vdi gia thiet. V?iy tat ca cac Xj (i = 1 , 2 , . . . , n - 1) deu khac
suy ra x„ diTdc xac dinh.
Til ket qua da ch\ing minh d tren suy ra tat ca cac x„ diidc xac dinh khi vk
chi khi xi ^ <„,Vn = 1 , 2 , 3 , . . .
Liru y. T a c6 the cluing minh / la song anh ngSn gpn hdn nhu sau: Vdi mpi

.

x,yeR,x^

...
thi

c
y =

Den day ta gftp man thuan. Vay: x = - — — ^
" '
cy - a

Vfc = l , 2 , . . . , n - 1 .

/ ( x i ) =
Khi do:

. i
b-dy
-d
, i b-dy
-d
De thay x =
^ — vi neu
= —
cy - a
c
cy - a
c

) = > / ( X 2 ) = X3 = ti

Gia sir xi 7^
Vfc = 1,2,. . . , n - 1. T a can chiing minh x„ dUdc xac
dinh. Do xi 7^ ti nen X2 dUdc xac dinh. T a c6 X2 ^ ti vi neu X2 = h thi:

T a gia sii phan chiing rkng / ( x i ) = / ( i j ) .

ax + b
,
.
u=
: ^ cyx + dy = ax + b'^x=
^
cx + d
"

suy ra X4 khong xac dinh, mau thuan. Vay x i ^ ^3. Tudng tu, suy ra:

TrUdc het cluing minh / la ddn anh: Gia suf x i , X 2 e R\| va xi ^ X2
ta can chilrng minh / ( x i )

X2 = t2

--,y^-,t'Ac6
c
ax + b
— J

cx + d

,
,
^ cyx + dy = ax + b-^x=

b ~ dv
-.
cy-a

/ la spng anh.
•^''0 i^n) la day phan tuyen tinh nhu sau

—.
c

T6ml?iiv6imQiy G R\)

ton t a i x = ^ - ^ ^ e R\ — | sao c h o / ( x ) =
(c J
cy — a
l " J
Hay la / la toan anh. V i / la ddn anh va / la toan anh nen / la song anh
6) T a se cluing minh rftng x„ (n > 2) ditdc xac dinh neu va chi neu:
XI
(=*•) G i a

Slit

x„

(n

Vfc = l , 2 , . . . , n - 1 .

> 2) ditdc xac dinh. K h i do vdi mpi fc = 1,2,...,

Xk dudc xac dinh va Xk^h.
Xl=

T a c6 xi 9^ ti. Neu x i = t2 thi:

r\h)=>

f{xi)=^ti^X2

356

= ti

n -

1

cIo ta CO cac dinh If sau day.
^ m h l y 2 0 . Neu day { x „ } hoi tu den L thi cL"^ + {d-a)L-b
^•^'J^g m i n h . T I T X „ + I = " ' ^ " " ^ ^ , V n = 1 , 2 , . . . cho n
cx„ + d
aL + b
^2
L = ——
^ cL' + {d-a)L-b
cL + d
^
'
357

= Q.

= 0.

+ 0 0 ta dUdc

D i n h l y 2 1 . Neu A = ( d - a)^ + 46c < 0 thi day phan

ki {khong

hqi ty).

D j n h l y 2 2 . Gid sU A = {d - af + 46c > 0 . GQI a,P Id hai nghi\$m
phuang

CUQ,

{dn Id x) cx^ + {d - a)x - 6 = 0. Khi do:

tfinh

6c)(xn -

0)

=

a) £i = a khi vd chi khi x „ = Q , V n = 1 , 2 , . . .
6) Old thiit

X I 7^ a, dat Xn = p—^yn
x„ - a
Xn+i

c) Gid thiet

G N*, A =

^

cP + d

. Khi do:

• Neu |A| =

= AX„, Vn = 1 , 2 , . . .

n—'+oo

l i m x „ = (3.

_

n—•+00

Neu

|A| =

Xn =

lim

ca + d

c.(i +

d

• Neu A = - 1 vd xi

AX„,Vn=l,2,...

n—.+00

< 1 thi

> 1 vd x i

-J-

TCr Xn =

/3

7^

ky.

• Trudng hap A = 1 A;/iong
• N e u |A| > 1 t h i

,i

C h u f n g m i n h . V i o, 0 la iighi^ni ciia phirong t r i n h L = " f ' ^ nen
cL + d
,

—„0 =

ca + d

lim

Xn

^

^

^

^

A „

-

1

^

lim

~ ^ =

lim

x „ =

n — + 0 0

= 0 ( d o Xi

lim

=

=

lim

~ ^ = /?

n-.+oo

Xn

1

0). D o do

lim

a - i -

,

_ Q- 0

a0 + b

l a i l a h i e n n h i e n . T a d i m g p h u o n g p h a p q u y n a p . G i a s i i x i = a. K h i d o
axi

+ 6

aa + b

cxi

+ d

ca + d

: =

t a CO

cp + d'

a) T a c h i c a n c h i i n g m i n h n e u xi = a t h i x „ = a , V n = 1 , 2 , . . . v i c h i l u ngUdc

X2 =

= 0.

0.
Q

X,J:n--V„.Q = x „ - , 3 ^ x „

• yVeu A = - 1 vd xi ^ f3 thi day { x „ } phan

aa + b

~~

X„ -

l i m X'^'^Xi
n—+OC

lim x„ = a.

P thi l i m x „ = (3.

a =

~ a)

c) T h e o k i t q u a c a n 6), ,suy r a A",, = A " - ' X i , V n = 1 , 2 , . . .
• N e u |A| < 1 t h i l i m A " " ' = 0. D o do

xi ^ a
CQ + d

cQ + d

(•0 + d
cn + d x „ - B

J

=

• N e u A = - 1 v a x i =^9 t h i x „ = / 3 , V n G N * s u y r a

• Neu A = - 1 \h,xxi-

7^ 0 v a

0 thi
X„+i

a.

l i m x „ = ^fl.

'

n—'+00

i,

=(-l)"Xi,Vn=l,2,...

Ta se c l u i n g m i n h d a y so ( y „ ) v d i y „ = ( - l ) " , V n = 1 , 2 , . . . , k h o n g h o i t u
(phan k y ) .

G i a siif x „ = a. K h i d o
axn + b

an + b

cx„ + a

ca + d

Cach

.

1 . T a c6

phan ky.

lim y i n - x
n—>+oo

=

h m (-1)
n—'+00

=

- 1 / 1 =

l i m 2/2n- V a y
n—'+oo

day

Vay theo nguyen l y q u y n a p suy r a neu x i = a t h i x „ = a, V n = 1 , 2 , . . .

C a c h 2 . V d i m o i a thuQC K t a c l u i n g m i n h d a y (?/„) k h o n g h p i t u v e a. N e u

6) T a

" 7^ 1 t h i |a - 1| = (5 > 0. C h o n e = ^ |a -

CO

Xn+l

=

Xn+i

-

0

(aXn

Xn+i - a

+b

\cxn + d

a0 + b\

+b

''0 > n sao c h o y„u = 1, n g h i a l a

aa + b

c0 + dJ ' \cxn + d

k h i do Vn = 1,2,... ton tai

ca + d

l2/no -a\

\\-a\>z.

ac0Xn + adxn + bc0 + bd- ac/3x„ - ad0 - 6cx„ - 6d
{cxn + d){c0 + d)
/ acaxn
V

+ adxn + baa + bd- a c a x „ - ada - 6cx„ - 6d^
icxn + d){ca + d)
358

,

" o n e n d a y n a y k h o n g t h e h o i t u ve a. N e u a- \i a 7^ - 1 . L y l u a n t i t o n g
V u h u t r e n t a c u n g t h a y d a y (i/,,) k h o n g t h e h o i t u v e a. V a y d a y ( y „ ) k h o n g
(2/n) k h o n g h o i t y m a A : „ + I = 2/„Xi, V n = 1, 2 , . . . v a
359

7^ 0 n e n d a y

X

—3

{Xn} cung khong hoi t u . Vay t i l X „ =

2 c ( x n - g)

(a + d)(x„ - g)
(a + d ) '
2c

(a + d ) ( x „ - g )

suy r a day { x „ } kh6ng hpi

G R t h i l i m X„ =

(vi neu h m x „ =

_ 2 c ( x n - g) + 2cff + 2d ^

2c
a+

a + d'^ ia + d){x„-g)

, dieu nay mau thuan v6i d a v

2{cg + d)

^

(a + d ) ( x „ - 5 )

1

d+
'^x„-g

I2(c9 + d) = a + d VI 2cg + 2d = a - d + 2d = a + d).
\u x i = g t h i theo a) suy ra x„ = g, Vn = 1,2,... do do

{X„} khong hoi t u ) .
• Tritdng hop A = 1 khong the xay r a bdi v i n^u A = 1 t h i
^ = 1
ca + d =' ciJ + d -t^ ca = cf3
a = (i.
cp + d

C) ^

l i m x„ = g.
n—•+00

fjlu X I 7^ g t h i theo b) t a c6 Xn+i = Xn +
Vn = 1,2,... suy r a {X^}
c^p so cpng C O cong sai la f.i va so hang dau la Xi. Do do

Dion nay khong thfi xay r a dUdc do A = (d - 6)^ + 46c > 0.

Ih

X „ = X i + ( n - l ) / y , V n = 1,2,...
D i n h l y 2 3 . Gid thiit A = {d - a)^ + 46c = 0 ua d(it g =

Khi do

2c

a) x\ g khi vd chi khi x„ = 5, Vn = 1,2,...
6) Gia thiet xx ^ g, ddt X,, = — - — , Vn = 1,2,dat
Xn- g
X „ + i = X „ + M-Vn = l , 2 , . . .

=

Khi do
a+d

lim

n->+c»

Chufng m i n h .
a ) V i A = 0 n e u phUdng trinh cL^ + {d - a)L - b = 0 (tutc l a phuong trinh
^ ) C O nghi^ni kep l a y = ^ T T - ^ . Tiep theo t a l a m tUrtng tit nhir da

L =

lam d dinh ly 22a) d trang 358.
6) Vdi moi n = 1, 2 , . . . , t a c6
1
Xn+l

Xn+l

-9

_
— i^

_

faxn

+ b

' \cXn

+

a

-d\
2C

d

'•

J

2 c ( c X n + d)

2acx„ + 26c - acx„ + cdx„ - ad +
^

2c{cx„ + d)
c(a + d)xn + 26c - ad + d'^'

V\A

= {d-

li iium
i —

=
=

n->+oo A „

a f + 46c = 0, nen 26c =

26c - ad + d2 = -

^±1^

=

i

- a-") =-\{a

;

= u
0
-r— =

(U — l)/i

l i m x„ = g.

Cac b a i toan.

Nhir vay t a da khao sat day phan tuyen t i n h , da t r a Idi diXdc cau hoi v6i dieu
ki?ii nao ciiia a, 6, c, d t h i day phan tuyen t i n h hoi t y , phan ky. T u y nhien ket
qua cua dinh l i 2 2 ci trang 358 va 23 ct trang 3G0 k h i d i t h i neu s\t dung t h i
phai trinh bay luon ca phan chiing m i n h cac ket qua do. Do vay neu t a nhd
dudc cac dinh ly va cac chiing m i n h ciia cac dinh do t h i t a c6 thg giai dufdc
hau het cac bai toan ve xet sir hoi t u ciia day phan tuyen t i n h . Sau day la
cfir bai toan ve xac dinh day phan tuyen t i n h , xet sir hoi t u cua day phan
tuyen tinh. D l xet sii hoi t u t a chi can ap dung dinh ly 2 2 , 23 la r a ket qua.
Tuy nhien t a ciing c6 the giai b a n g each khac, mac d u giai b a n g each khac
" l i H i g cac kot qua t l cac d i n h ly do c6 tac dvmg d i n h hucing, giup t a t i m dugc
' ^ i giai nhanh chong hdn.

=

2 ^ ' " ^ ' -

- d){a + d) = - i . 2 5 c ( a + d) = - c ( a + d)g.
^^<3ng d a n . Xet hai day so (a„) va (6„) nhit sau

Tit do
^
,1^

2c(cxn + d)
""^^

c{a + d)x„ - c(a + d)<7

^

2c(cxn + d)
c{a + d){x„-g)
360

liiim
i u x„ = g.
n—+00

fiai t o a n 2 4 4 . Ttm so hQng tdng qudt cua day so (u„) cho nhu sau

Do do

2.10.4

l ^ - '
J-{d^

liim
mi —
n-.+oo A i +

V^y trong moi trUtJng hup t a dcu c6

Um Xn = 9-

c)

(x„ - g) =

^

2(cx„ + d)
{a +

d){x„-g)

ai = 6i = 1
a„+i = a„ + 26„, n > 1
6„+i = 2a„ + 3 6 „ , n > 1.
361

C a c h 2. Ta chutng niinh
V<3i inoi n = 1, 2 , . . . t a c6

_^ ,

(2)

a„+2 = o„+i + 2b„+i = a„+i + 2 (2a„ + 36,,)
= a„+i + 4a„ + 3 (26„) = a„+i + 4a„ + 3 (a„+i - a „ )
= 4a„+i + a„.

Ta CO I'n = 1 >

- 3 + %/5

. Gia svtvn

Q,
^.^^-3 + ^
3 + v„ > 3 H

V?Ly day ( a „ ) thoa man diou kicn

>

K h i do

3 + v/5
= — z — =>

2

ai = l,a2 = 3; a„+2 - 4a„+i - a„ = 0, n > 1.

Bcii vay
Vn+\ -—^—

1

3 + i;„ - 3 +

3 + u „ - 3 + \/5

T u o n g ti^, so h^ni^ t6ng quat ciia day (6„) la:

2

<

- 3 + 75

-2

>

^

2

Theo nguyen ly quy n?ip suy ra (2) dung. Do do day [vn) b i chan dir6i. Ta c6
^

1

vl + Zvn + \ - 3 + \/5,

3 + t;„

3 + Vn

lira Vn — L.

Vay day da cho giam va bi chan dudi nen hoi t u . G i a sii

n—>+oo

Vay: u„ = 7— =

7^^

7=

> 1.

1,2,...

Cach 3. Dat L =

k l i i do L =

ClivCng viinh rdiiy day do. cho c6 yidi han hSu haii va thn gidi lian do.
Giai.
C a c h 1. ( v „ ) la day phan tuyen ti'nh, v„+i = — - — 3 , vdi a = 0, 6 =
CVn + a
c = I vk d = 3 {ad - be = I

0, c

0). Phuong t r i n h

nghi?m plian bi^t la x\ p

va X2 = a —

1

1

3 + L

3 + Vn

(3 + L)(3 + t;„) >

n—+00

v„^a=

" \$ 1^. - L| < ( - j

- 3 + v/5
2

.

Ta cd

\Vn - L\
(3 + L ) ( 3 + i;„)"

/3jV5\
2 j "'4-.

I

Do do
A
/
l^n+i - L| < - |t;„ - L| < / - )

hm

3 + L"

_LlL^_
2

Uk-\4.T
3+^5
^ - 3 + \/5 ^
.
^Ha J + L = — - — , Vn >
, Vn e N nen
Z
It

+ 3x + 1 = 0 c6
. T a co

Vay theo diuh ly 22c), day d a cho hoi t u va

+ 0 0 t a dime

(3). T i t (2) cho n

'^^"^

L = -—Giai
phudng t r i n h nay vdi dieu kien (3) t a d U d c L =
o 4" X/
,.
- 3 + V5
V^y hm v„ =
.
n—+00
2

B a i t o a n 2 4 5 . A'e< day {v„) xdc dinh bdi VQ = 1 va
vn = — ^ , V n =
3 + i;„-i

Khi do tit (1) .suy ra L >

' ^ y ^ ' ^ ly kep t a d U d c

f A
|t;„_i - L| < • • • < ( -

|wo - L| , V n = 1, 2 , . . . cho n
lim

^„ = L

n—*-\-oo

=
2.

\

\v^-L\.

+ 0 0 va sU dung

Bai

t o a n 246. Cho day { r „ } nhu sau: vi = a va

Ta CO 6 + cvn > b + cv =

thtJc fix)

. ^, = -

,Vn 6

(a, 6, c > 0, A = 62 - 4 a c > 0, a >

(1)

> 0. Yi v„ > V, vdi V la nghi^m Idn ciia tam

= cx^ + bx + a nen cv^ + bvn + a > 0. Do do v„ - v„+i

-6+
C/i»?np

m i n / i rdng

s6

day

da. cho

hoi tu va tinh

hpi tu. Tir r i

^}^rn^^n-

lim
Giai.
Ckch 1.
1. N§u
Neu

lim i;„ = L tlii tit (1) cho n -

> v 2 > V 3 > - > v „ >

Vn > — — V ^ y

+oo ta dUdc

lim

K y hi§u

= —

T

cluing minh

'

b + c

< cv^ ^

cv.Vk-i

+ bv + a < cv^ + bv +

T a CO 6 + cvk-i

>b

+ cv,

ma

v =

b + cvk-i

+ bv + a <

a.

2c

6 -

v/A

6 -

6 + v/A

< 1.

6+v/A

2c

v/A

lim t;„ =

.
ZC

n—'+00

Bai toan 247 (De thi vo dich sinh vien Moskva-1982). Cho day {x„}
sau: xo = 1982, x,.+i = .

0.

\n = 0 , 1 , . . . ) . Hay tlm

4 —0 X „

lim x „ .
n->+oo

ueu
+ cv =

CSch 1. T a CO ( x „ ) la day phan tuyen tinh, x „ + i =

, vdi a = 0, 6 = 1,
cx„ + a

>0.

c ^ - 3 va d = 4 (od - 6c = 3

0, c ^ 0). Phudng triuh S i ^ - 4x + 1 = 0 c6

''ai nghi^m phan bi?t la xi = ;3 = ^ va X 2 = Q = 1. T a c6
+ bv + a

b + c.Vk-i

a

—a

b + c.Vk-i

b +

i

c.Vk-i

Vay (2) diing khi n = k. Do do theo nguyen ly quy nap suy ra (1) dung ^''^
nioi n = 1 , 2 , . . . Vay ta da dn'nig minh dttrir day so da cho bi chan difcii T-'*

theo djuh ly 22c) suy ra day da cho hpi tu va ^ h ^ ^ v„ = 0 = ^•
2. Xet ham s6 / ( x ) =

CO
a
Vn - V„+i

nhu

Giai.

Boi vay
c.v.Vk—i

2c

-b+

—a

>b

2c

2c

Vay theo (hull Iv 22c) suy ra

Vi V la nghi?ni ciia piiUOng triuh cx^ + bx + a = 0 nen cv'^ + bv + a = 0. Do do
cv.Vk-i

^/A

+

v/A

- 6 -

b +

^ s u y
ra (2) dung khi n = 1. G i a sijf (2) dung tdi
2c
n = A; - 1, tiic la Vk-i > v. K h i do t;(i;jfe_i - f ) < 0 (vi i; < 0). Do do
cv.Vk-i

, Vn = 1, 2 , . . . , suy ra

Taco
(2)

Tir gia thiet Q >

<

-b

Vn =

2c

- 6 + v/A

v.Vk-i

V/A

C a c h 2. T a c6

2c

a

>

n—»+oo

- 6 ±

6 + cZ,

v„+i

theo \i lu(ln d phan d i u Idi giai suy ra

n->+oo
-a

> 0, vdi

iiipi n e N * . Hay (i;„) la day giam. V?iy (t;„) la day giam va bj chan dudi nen

= Vn + -r-,
=
b + CVn

364

cvf. + bvn +
7b+

CVn

khi do fix)

lien tyc tren R \| va

a

x„+i=/(x„),Vn = 0,l,...
365

fix)

T a CO

=

(4 - 3x)

fix)

=x^

1

=

4-3x

3x^ - 4x + 1 = 0 <=>

X

phirffiig t r i n h dac t n m g cua day so {y„} la

rx = i

So h * " g tSng

"=3-

1

-1

4 - 3.1982

5942

''''

1^-

Vi zn =
Vi j/i =

.0,24997.

2n = >1 + B . 3 " + ' , Vu =: 1 , 2 , . . . V i 21 = 1 nen 1 =
XI

nen xi=

A+

35.

V§y: | ^ +

A =

f f ^ <^

X2 <

4
<
4

3x1 - 1

1

2

nhiii vao bang bieu thien

Ta

^

ta thay iigay /(x) la ham

'

so tang t i t ctoan [0; 1] vao

•^'"^

= X , .

CO

1

hay

6

^ ^ \n = 3 , 4 , . . . Suy ra day (x„) hoi t u . D a t
4'3
va

4'3

X

-.3"]
2
+
6
3x1 - 1
T^I^T
' Vn = 1, 2 , . . .
- ^ ± . - ^ . 3 - ^

G

l i m x „ = x,

Sl^r^^3T(r31^;)3^'^" = l ' 2 , . . ^

n—>+oo

'Jo Xn khong xac: dinh khi va chi k h i :

Vay t a lo9.i tritdng h0p x = 1 . Do do x =

- 3 + ( 1 - ^ J 3 n + i = 0 ^ ( 9 - 3 " + i ) x i = 3 - 3 " + ' ^x,

l i m x„ = - .
n-'+oo

1

4- 3x;

3 x ^

t r d d i , day (x„) la day so tang.

X

X,

Vay theo nguyen l y quy nap suy ra

Do do thoo kot qua bai toan 50a) 6 trang 148, suy ra ke t i t so hang t l n i hai

khi do

1 -

6

-3t/„ + 4z„ " 4 - 3 ^

= /(X2) > ^ = 0 , 2 5 > X 2 .

1

3x, - 1

Gia sit ~ = x„, k h i do

^n+i

O

1

- x i
6

Vn+l

(loan [0; 1], va - > X 3

De thay x„ e

1

B a i t o a n 248'. Cho day so { x „ } dinh nyhta tr~uy hoi bdi:

%

4-3x,

Hay tlm cdc gid tri cua x i dc day trcn hoi tji vd trong cdc trUdng help do
tinh l i m x „ .

ta CO k^t qua nlur sau:
'

. xrl:

* ^"^i c a c

^

3 - "3 ^ '
3
" ^
'

1

day khong xac dinh.

^ " = 1. Vn = 1, 2 , . . . , do do

l i m x „ = 1.

^

9x1-3.+ (l-xi)3"

^, vdi a = 0, 6 = 1 , c = - 3 , d^^'
CXn

+

d

0. Xet hai day so (y„) va {z„) thoa man dieu kien saU„lim

yi

"-+00

366

'

k; ciia x i t h i x„ xac din]
gia t r , khac
dinh vdi moi n = 1 , 2 , . . . v^

n—>+oo

GTiai. Day so da cho c6 dang x„+i =

LJ1_

^ 1 - 3 "

• Khi X

(n=l,2,...).

=

3-3«

i

c ^ 0, ad - be = 3

3 x ^

B =

Do do vnii moi n = 1 , 2 , . . . t a c o :
Vay 0 <

A= 1
A = 3.

y„ = >1 + S.3", Vn = 1 , 2 , . . . (^, B la cac liKng so se t i m sau).

Taco
x i = / { x o ) = /(1982) =

c"^*

- 4A + 3 = 0

x„=

Ihn
n^+^

-g^-3+(l-xi)3"
9a;i - 3 + (1 - a:i )3"+i
367

+ 9B.

911 - 3
=

3x„+i - 1

lim

„"+oo 9 £ | ^ ^

_ ^^^3

(l-xi)3

3"

•3a:„ _ 1

3 " - ' ( x , - 1)

"

'3x„_j - 1

„„_i

Xi -

'3x1 -

1

r

,

Ltfu y.
De cho IcJi giai dUdc ngin gpn va hdp logic thi vi§c tim ra cong thiic
t6ng quat x„ cua day {a;„} diWc lam d ngoai giay nhap, con khi trinh
bay 16i giai ta tien hanh theo trinh tif sau:
- Bang quy nap (theo n G N*), hay chuTng minh: u„ dvr0c xac dinh
neu va chi neu: (3* - 3)xi
3* - 1 khi 1 < A: < n;
(1)
hPn nuta, vcii dieu ki^n (1), thi
( 3 " - ! _ 1) _ (3"-! _ 3)xi
(2)

(3" - 1) - (3" - 3)xi
- V^y, Hipi so h ^ g cua day deu dM0c xac djnh neu

chi n6u:

3x, - 1

^-1

g a i toan 249 (Do thi vo dich Kicp). Cho day so (a„) dU(fc xac dinh nhu
sau:
3
ai = 2 , a„+i = 4

(Vn = 1 , 2 , . . . ) .
a„
Chttriy minh rdny day ad da cho c6 gidi h(f.n hHu hg.n va tinh gidi h^n cua
day so (i^Dap

so.

lim a„ = 3.

Bai toan 250 (De thi hpc sinh gioi qupc gia, bang B, n i m hoc 2002-2003).
Cho so thuc a ^ 0 va day no thxjtc { x „ } , n = 1 , 2 , 3 , . . . , xac dinh bdi:

3 ^ , ^ > 2 } .

(3)

Ngoai ra, vdi digu kien (3), thi (2) diing vdi mpi n e N*.

I

- Tijr nay, gia suf (3) dit0c thoa man. Khi do, neu a 7^ 1, de thay: (2)
keo theo
ton tai ciia gidi han:

lim

n—'+00

x,i —

lim --,
I _
3-i

1
m-l

^
1 \

-

- Con neu o = 1, thi x„ = 1 (Vn € N')

1 -a
3-3a

1

3 - m-2
lim

Xn =

3

x„ = 1, Vn = 1, 2 , . . . , nlu x, = ^ thi x„ = \ Vn = 1 , 2 , . . . T i l p th^'"'

IViidng hdp 1: a = - 1 . K h i do x „ = 0, Vn = 1 , 2 , . . .
IVUcJng hdp 2: a

- 1 . K h i do x„ ^ - a , Vn = 1 , 2 , . . . Do do ta c6
x„+i = ^ ^ , V n = l , 2 , . . .
x„ + a

1
3(:^»-l)
- 1 =
4-3x„
4-3x„
3x„-l
3
- 1 =
=
4-3x„
4-3x„
368

(1)

(1) CO dang x,.+i =
vdi a = 0, 6 = a + 1, c = 1, d = a, c ^ 0 va
cxn + d
"'^ = a + 1 ^ 0 ( a 7^ - 1 ) . Xet hai day so (j/„) va {z„) thoa man dieu kien

'^J'i do

x„+i - 1 =
3x„+i - 1

b) Chiing minh day s6 (x„) co gidi han hHu hQ,n khi n —> +00. Hay tim gidi
ban do.

1.

Ngoai ra ta c6 the tim dUdc so hang t6ng quat cua day bang pli'^'^"^
phap ham l^p (xem myc 1.1.5 ci trang 24) nhu sau: Neu x i = 1
1
xet xi 7.^ 1 v a x i 7^ - . T a c6

a) Ildy lim so hung long qudl cua day so dd cho.

Giai.
a)

1
in-2

xi = 0, x„+i (x„ + a) = a + 1 (Vn = 1 , 2 , . . . ) .

J/„+2 = (a + 1) zn+i = (a + 1) (y„ +

369

az„)

.

f lidng t u tritcJng hop 2a, bang quy nap ta clnbig niiuh difdc:

Vay plntdiig t r i n h d i e t n m g c u a day so {yn} l a
-

QA

y^_{a+l)
( - 1 ) " + (g + 1)" _ (g + 1) [ ( - ! ) " + (g + I f i ]
^"==z„
(-ir+'+(g+ir
+
-VneN*.

A= -1
A = a + 1.

- (a + 1) = 0 <^

Trvrdng hdp 2a: a = - 2 . Khi do phUdng trinh A^ - aA - (a + 1) = 0 c6
nghiem kep Ai = A 2 = - 1 . Suy r a

y^ = {C + Dn) ( - 1 ) " , Vn = 1 , 2 , . . ..{vdi C va £> la cac hang so se t i m sau).
Vi

2/1

= 0 nen C + £> = 0. Vi j/2 = (a + 1) zi =

V a y { g t S f i _ i ^{g=i:^
y„ =

Q +

1 = - 1 nen C + 2£) =

J) Theo k i t qua cau a) suy ra:
. Neu a = - 1 thi x„ = 0, Vn = 1 , 2 , . . . Do do

n - 1
, Neu a = - 2 thi x„ =

-1

lim

n—'+00

—- =
n

n-*+oo

lim ( 1 - i )

, Neu g 7^ - 2 thi
,Vn=

Ta CO — = x i . Gia Slit — = x„, khi do

ia+l)zn

a+l

_ Q+ 1 _ „

1.

n->+oo \

Q + 1) [ ( - l ) " + ( a + l ) n-l

y„+i

'••'ax

Vn = 1, 2 , . . . Do do

lim Xn -

Dodo

(l-n)(-l)^Vn=l,2,...,^„=|^=n(-lr^Vn=l,2,...

lim x„ = 0.

1,2,...

Taco

,

Zn
V\$,y theo nguyen ly q u y nap s u y ra

;\

x„ = — =
Zn

,Vn=

1,2,...

n

Trtrdng hdp 2b: n ^ - 2 . Khi do so hang tong quat ciia day so {y„} la:

yn = A ( - 1 ) " + B.{a + 1)" , Vn = 1,2,...{A,B

la cac hang so se t i m sau).

Vi 2/1 - 0 nen -A + B. {a + I) = 0. Vi y2 = {a + 1) zi = a + I nen t a c6
A + B. (a + lf =n + l. Vay
-A + B.{a + l) = 0
A + B.{a'+lf
= a+ l

{

A -

2^271

-

=

+ 1 + (» + 1)^"
(g +

_

(Q +

l)'"-l

1 -

\2n1)

+1
-,Vn-

1,2,...

,2n

(g+1)

•Do do ngu | Q + 1| > 1 thi
lim

X2n =

n->+oo

lim X2„-i = 1 =>
n—+00

lim x„ = 1.
n-»+oo

•Neu | Q + I| < 1 thi
Q +

2

lim x 2 „ = - ( g + l ) =

lim X2„-i

lim x„ = - ( a + l ) .

Trong so hang tong quat ciia day so d bai toan 250 0 trang 369 lay

V\$,y vdi mpi n = 1 , 2 , . . . ta c6

a + 2

(g + 1)

~3 ta dUdc so hang t5ng quat cua day so trong bai toan 247 ci trang 365
a+2

^ ^ ^ g khac gia tri ban dau. Ta cung c6 the tim dUdc so hang t6ng quat ciia
y so nhanh hdn bang phudng phap lap, diia tren cac ket qua sau:
^n+i - 1 =
Xn + a

370

_ 1 = -^" + 1 ^ - {Xn - 1)

x„ +a

x„+a
371

ia + l)x„ + (a + lf
X n + l + « + 1 = Xn
x„+i+a+l

+—a + " + 1 =

a + lXn

^ ( g + 1) (a:,. + a + 1)

x„ + a

+ a+ 1

x„ + a

p a l t o a n 2 5 3 . Cho day so (u„) n/ii/ sau UQ = 0 vd

( a + 1 ) " ' x i + a + 1'

- u „ -H2010'
B a i t o a n 2 5 1 . Cho day so {x„} nhu sau:

fl) Chiing minh day {un) c6 gidi han hHu han vd tinh l i m u„
1

'')^^^^" = £^;;^2oo8- ^^''^^ n-^+oo
Chiing minh rdng day {xn} hqi ty. vd tim Urn Xn-

G
i j ii aa ii ..
^
a) Ta CO uo < 1. Gia siif u„ < 1, k h i do do ham / ( x ) =

H\Xdng d a n . D ^ t (\/2 + l ) " " ^ ' x„ = u„. K h i do
do
UQ = 1, u„+i =

hm

Xn

=

^

^ ^, Vn = 0 , 1 , 2 , . . .
3

2008
Theo nguyen l i quy nap suy ra u„ < 1, Vn € N . L ^ i c6 u j = ^QJQ > UQ. G i a
sii

= \/2 - 1.

u„ > u„_i =^ / ( u „ ) > / ( u „ - i ) =^ u„+i > u „ .

B a i t o a n 252 {D^ t h i HSG Gia Lai, nam hoc 2008-2009). Cho day so (x,,)
(n

0 , 1 , 2 , . . . ) thoa man

Theo nguyen l i quy nap suy r a u„+i > u„, Vn e N . Vay day (u„) tang va b i
chan tren nen c6 gicii han hfm han. Dat
cho n -+ + 0 0 t a dUdc

L =

i

l

—L
lim
n->+oo

6) Ta

x„.

b) Chiing minh x i + X2 + • • • + X2008 < 2009.
H t f d n g d i n . Sii d u n g

1.1.5 6 t r a n g 24)

de d a n g t i m d u o c so h a n g tong q u a t c i i a d a y so ( x „ ) l a x „ =

He

2
3„+i

_ i = l + i + 3+

1
32 + . . . + 3 n < l

^

^

L

2

-

2009L + 2 0 0 8 - 0 ' ° ^ ^ ' L = l

+

1

y^+rZi'

1
3;r<

n—>+oo

- 2008 = ^009

^

-Ufc-i

Ufc - 2008 ~ 2009

, J_
+2^-

^ l i m u„ = 1.

CO

-Uit-i+2010

ta

^ I

Uk - 2008,

2009

-2008)

-Ufc_i+2010

-h2010

2

1_

- 2008) ~ 2009 {uk-i - 2008)

E

n

l

^ ^ j U f c _ i - 2008

2009'

Suy ra

quala

n;u€

±

+ 2010

_ 2008 =

phiXdng p h a p h a m l a p ( x e m m u c

3"+i

= 1+

l i m u „ = L , k h i do L < 1. Tilt (1)
n-.+oo

2x„ + l
xo = 2,x„+i = ^ : ^ , V n = 0 , l , 2 , . . .
a) Tim

dong bien

ti„+i = / K ) < / ( ! ) = ! .

r ; i + r , Vn = 0 , 1 , 2 , . . . Vay
- ( v 2 - 1)

Hm

2008

X ~T~ ^UXU

tren khoang ( - 0 0 ; 2010) nen

u„ +

Ta chi'mg m i n h ditdc u„ = — ^

(V2 + 1)

n + 2009 •

+ ^ - f . . . + ^ = n - H - ^ < n +l .

fc=i

"

uo - 2008

f^^Uk-

Uo - 2008

2009 £ ^ u ^ - i - 2008

2008

n

Vay t a c6 dieu phai chiing minh. Hdn niia t a t h u dUdc

Lfc=l

= n.

372
373

2009

2009

-1
2008

+

2

n

Tn

2009

Un - 2008

o„-i

-K

-

cos a
(a„_i cos 2a + 1) (cos a - cos 2a) cos^a'

2009'

if)

\u

TifOng tir

V^y
2007
_
2
2009 " ~ 2008
2009(u„ - 2008)
-2009
'
2
^Tn =
2007.2008
2007(w„ - 2008)

1

n
2009
n
2007'

a„ +

cosa
(a„_i cos 2a + 1) (cos a + cos 2a) cos'^a'

cos a

(2)

rp^ (1) va (2) suy ra
( c o s a ) a n - 1 _ [ ( c o s a ) a n - i - 1] (cosa - cos2a)

Vi
n
= 0, l i m
n
"
+
0
O
2007(n
+ 2009)
2007.2008(71 + 2009)

n^+oo

2
n "lini
+oo

(cos a ) a „ + 1

1

-2009

[(cos a ) a „ _ i + 1] (cos a + cos2a)
_ [(cosa)a„_2 - 1] (cosa - cos2a)^

2007

[(cos a ) a „ _ 2 + 1] (cos a + cos 2a)^

= 0

2007(u„ - 2008) (n + 2009)

_

_ [(cos(t)ao - 1] (cosa - cos2a)"
[(cos a)ao + 1] (cos a + cos 2 a ) "
(cosa - cos2a)"+^
(cos a + cos 2a) n+i-

L v f u y. Ta cou c6 the giai bang each t i m so hang tong quat

\

2 ^n
V2009;

n

-

1

2008 V2009y

- 1

2.11

D a y tong
M o t s 6 \\iu y.

dc siiy ra kct qua.

2.11.1

B a i t o a n 2 5 4 (De nghi Olympic 30/04/2011). Cho day (a„) xac dinh bdi:

Dinh n g h i a 21. Cho day so (x„)+ri- Xet day s6 (5„)+^i nhu sau:

1
°"~cos2a'''"

Un-i + 1 - tan^Q
a„_iCos2a +r

n

Tim so hang tong quat cua day va xac dinh gidi han cua day.

a„_i + 1 - t a n ^ Q

1

a„-icos2Q + l

cos a

O n - 1 cos a + cos a -

sm^Q
COSQ

- 1 - o„-icos2a

( a „ _ i c o s 2 n + l)coso:
cos a (cos ct - cos 2 a ) O n - i + (cos^a - sin^a - cos a )
(a„_i cos 2 a + l ) c o s 2 a
cos a (cos Q - cos 2 a ) a „ - i + (cos 2 a - cos a)
( a „ _ i cos2a + l ) c o s 2 a
374

Xi, tlic Id Sn = Xi + X2

i=l

do day SO {Sn)l=i

diMc got Id day tSng.

, vivn;

1- Xn.

Ltfu y. M o t so (lay so eon dime xay diftig thong qua nhfmg tong thoa man
aieu kien nao do. Chang han cac bai toan 31 (6 trang 127), 33 (d trang 128),
34 (6 trang 129), 35 {d trang 130), 36 (a trang 130)...

Hirdng d i n . Ta eo
1
a„" - cos a

E

( - 2 < " < 2 ) -

l'

^ h a n x e t 2 5 . Doi vdi nhxtng day so {xn) cddangxn+i
= / ( x i + X 2 + -• •+x„),
hang each dat Sn = xi + X2 + • • • + Xn, ta dua nc day so { 5 „ } + ^ i nhu sau
Si = Xi, Sn+i = Sn +
fiSn).
Ch' '
V
^^.V y 32. Van de xet sif hoi tu hay phan ki cua mot day tong diMc dc cap
di^^
chuang trinh toan cao cap, tuy nhien trong bdi nay ta chi xet van
. ^ ^0 cap han do Id tim gidi han cua mot so day tong thudng gap trong cac
hoc sinh gidi trung hoc pho thong.
375

2.11.2

P h i f d n g p h a p t i m gidi h a n c u a day

t6ng.

vay

Bdi

T a tlnrrJiig d u n g c a r phUrtng p h a p sau d a y :
• R i i t g p n hoSc t h n so h a n g t 6 n g q u a t c i i a day so ( 5 „ ) (day l a d a n g t o a n r§,t

„ii!?«, ("1

+

"2 +

•• +

u„)

=

1 -

lim

t h u d n g g a p t r o n g cac k j ' t h i H S G ) .
• So s a n i i d a y so (sit d u n g n g u y e n l y k ^ p ) .

g a i t o a n 2 5 6 . Xet

• Sit ( i i i n g cac d a y c o n ke n h a u .
• C h u y e n ve d a y t i c h .

1)2

= 1.

w
, .
'''
(2n-l)2(2n+l)2'^"=l-2,...

Yl

lim

fim

ddy ( x „ ) nhu sau: x „ =

(n+

• Sit d u n g k e t q u a c u a b a i t o d n 7 2 6 t r a n g 163, 7 3 6 t r a n g 163, nh|,n x e t 5
d t r a n g 163, nh§,n x e t 6 d t r a n g 164, b a i t o a n 2 0 0 a t r a n g 289.
C h u y 3 3 . Neu
d trany

cho bdi he thiic

379, bdi toan 2 6 1 0 trang

Chung

{x„)

day ( x „ )

tang

• Rut gon

minh

lim

vd khong
tong

truy

hoi {ching

380...) thi ta thudng

=

+oo

hofic

bi chdn

tren

hodc day ( x „ ) gidm

x„

5 „ , tii do tim

lim

lira

x„ =

-oo

G i a i . T a c6

han bdi toan

Idrn nhit
bdng each

vd khong

260

n

sau:

^"

chi ra day

bi ch^n

dudi.

(2ri-l)2(2«+l)2 -

5„.

( X l + X 2 + --

C h u y 3 4 . De' rut gon

^

tSng S„ =

thudng

bien

(2n-l)2

(2n + l ) 2 _

,Vn =

l,2,...

Do do

n->+oo
n

1
§

-f-x-„) =

i
8

1 (2n+l)2

Bdi va\

doi

i=l
n

^ l i m ^ ( x i + x 2 + ... + x„) = i

n

lim

8 n->+(xi

i=l

1=1

B a i t o a n 2 5 7 . Tinh
2.11.3

C a c bai

B a i t o a n 2 5 5 . Xet

toan.

1

-

1
(2n + l ) 2 _

n
- 1
l i m V x^. biet: x„ = I n
,Vn =
ji—+30
n-* + 1

8'
2,3,...

G i a i . T a c6

ddy (u„) nhu sau: u„ =
lim

(lii

+1X2

H

! , „ , V n 6 N*.
n^{n + 1)2

Tim
^

2^-1

, 3 ^ - 1

n ^ - l

,

Hu„).

u->+oo

G i a i . T a c6

^•2(A.•+l)2 -

Jt2

^

(fc4-l)2'^*""~

^ + 1

+U2

^ (iiilK^lt^ _
(^•+1)(A:2-A:+1)

1^0(16

D o do
Ul

2^ + i A 3 ^ + i y

+

„,

/ 1

h

= 1 -

1

376

+ ly
li^TT
- 1) [(^ + 1)^ - (fc + 1) + 1
(RTJOt^TI^T)

377

,

.

,

Suyra

lim

ft

fe^

1
1
2
n2
2
^ . ^ - U i = ^. V i ham s 6 / ( x ) = i .

=

Xk =

n->+oo ^—^
fc=2

= ln

In

lim
n—•+00

1

•+

B a i t o a n 258. Tinh Jjm^ ( ^ + I T 2 " r + 2 + 3

k{k + l)

= 2

fc3
AA
;

n->+oo

/J

1
G i a i . T a c6
1 + 2 + --- + A:

fc=2

=1.2

1+ 2+ 3

^ - ^ ) + .

'

1
an(n + 1 )

a

. V i vay

4;

= 2 -

+

- - + ' ' n - n + i y

lim | — + — H

Haytm

n +1

1
r +

--'

+

Giai. V i a > 1 va u„+i =

l +2+

--- + n

=

2-

lim

2

\

n+ 1

= 2.

B a i t o a n 259 (De nghi Olympic 30/04/2013). Cho a Id mot so thxfc

— n

2a

: Xn+l

_

i+axn+

,

/—X—T—•—:

yja^xl

E Xfc- Ttm ^]im^'5„-

T u i = — I
Un-l

\"1

Un

1

U2

Un

Xn

"1-1

1

+ 4ay„+i = a^ + 4a2/„ + 4a^ + 4av/a2 + 4 a y „ , Vn € N * .
378

< •••

(2)

^-—- = — - - ^ — —
1
Un (Un - 1)
1
1
, 1 . 1 : , ; . .
^ — =
U n " ^ U„
U n - l U„+l - 1

V^l-1

taco

V f ^ - - ^ l + . . . +

W2-iy^V'^2-l
1

Nhan hai ye ciia (1) v6i 4a va CQng hai ve vdi a^, t a ditoc:

• • • < u,, < Un+l

U i < U2 <

1—-,Vn=l,2,...
+l - 1

1

G i a i . T i r gia thiet t a c6: x „ > 0,Vn 6 N*. Dat — = y„. T i t coug thiic

Vn € N*.

,

+ (u„ - 1)^ , Vn e N* nen de dang suy r a

. 1
.
1
' • . .
" u „ + i - l
U„ - 1

(I

dinh day ( x „ ) , t a c6: j / i = 2d va

a =

+ — )•

Un+\

Vn€N*.

+ 4aa;„

fc=i

y„+i = y n + a+ ^a^+4ayn,

(1)

u„+i - 1 = u„ (u„ - 1) <^

dmvj

Un

Dat Sn^

^

a

Td (1) va (2) t a c6:

Id day ( x „ ) dMc xdc dinh bdi:
Xl

h m Sn = -.

n-^+oo

u„+i = u ^ - u „ + l , V n = l , 2 , . . .

+n

1 <

/

7-7)=^

\ +1 y

Bai toan 260. Cho so ihuc a > 1. Xet day so (u„) n/iU sau: u\ a vd

Do do
lim 1 1
^ 1
+ ,^
n-.+oo \+ 2

'

1 / 1 1
a \n
n+ 1

=^5„ = V a ; * : = - f l

k+l

' l+2+---

V3

u„ - 2an + a => 2/„ = a(n^ + n), Vn € N*

+ ••• +

1

/I

= ln2
3'

+l

1
1+ 2

VneN*.

Do Un > 0' Vn e N ' va a > 0, t a c6 (3) <4> u„+i = u„ + 2a, Vn s N*, hay (u„)
la niot cap so cpng c6 cong sai d = 2a va u i = 3a. Vay

/ n
y

=K+2a)2,

+

lien tuc tren (0; + 0 0 ) nen t a c6:
lim

u„ = i / a 2 + 4ay„, u„ > 0. T i f (2) suy ra:

\ /

1

U n + l - l ) - [ a - l

U 3 - l J ^ " ' ^
1

[ u n - l

Un+i-l)

\

Un+l-l)''^''-'^'^'---

oh

J^^ng h u p 1: Day (w„) b i c h t o tren. K h i do day (u„) tang va b i chan tren
gidi han hihi han, D a t l i m u„ = L. T i i (2) suy r a L > 1. T i f (1) cho
379

+ 0 O t a dugc L =

n

3

- L + 1 hay L = 1. Dieu nay mau thuaii v6i L > j

1

Vay tritdng hcJp 1 klioug the xay ra.
Trvfdng h d p 2: Day so (u„) khong bi chan tren. Theo chu y 12 d trang 147
ta CO

hm ' u „ = + 0 0

Tl—> + 00

( 1

hni

a-

a- 1
L i f u y. Day so (u„) c6 h? thiic truy hoi la u„+i = / ( u „ ) , vdi / ( x ) =
Ta t i m diem bat dong ciia ham so / bang each giai phirong t r i n h
j(x)

r

= X <^

- I+

1 =x

x^ - 2x +

+i

1

1

x„+i - 3

xn+i-a)

Xn+x - 3

' 2

iTOfir

bi ch\$,n tren

suy ra L > 5. T i t (1) cho n tien den + 0 0 t a duoc
L = ^{L^ ~ L +
1 <=> (x - 1)2 <(4- X = 1.

Dieu nay man thuSii vdi L > 5. Vay trudng hop 1 khong the xay ra.
Trifdng h d p 2: Day so (x„) khong bi chan Iron. K h i do
1

Do vay, trong 16i giai bai toan 260, t a se x6t u„+i - 1 dg suy ra

2

1
+ ( ^ , , _3
\X2 - 3
X3 - 3 ) " ^
1 _
1
, V n = 1,2,...

5 = X i < X2 < X3 < • • • < x,, < x„+i < • • •

1

hm 5„ =

/

1
-3
1

T r i f d n g h d p 1: Day (x„) bi chan tren. K h i do day (x„) tang
non CO gidi han hfm han. Dat h m Un = 1.1x1

1

1

= hm

.0 <

"2

n—'+00

Vay

Hm (u„+i - 1) = + 0 0 . Do do
n—>+oo

xi - 3

X2

-,Vn = l , 2 , . . .
hm

x„ = + 0 0 =4>

Hm x„+i = + 0 0 =^

1:

hm

=0

Tir day t a rut gpn dildc tong cSm t i n h gidi han.
B a i t o a n 261

( D ^ t h i chmh thi'tc O L Y M P I C 30/04/2006). Cho day (x„)j

Du do

iim

nhU sau: x i = 5 ud
x„+i =

nm _ hm 5Z X j +

^(4-^n

+

9),Vn=l,2,...

2

la day so tang, t\t do x „ > x i = 5, Vn = 1,2,... Tir (1) t a c6
5 (x„+i - 3) = x^ - x „ - 6 = (x„ - 3) (x„ + 2)
^ x „ +^i - 3 = ( x „ - 3 ) ( x „ +
^ 2) ( v i x „ > 5).

1

1

x„+i - 3

x„ - 3

^ X i +2

^

n-+oo ^2

^

=-.

X„+i - 3 /

1

, Vn e N ' , hay
x„ + 2

- 1 - = „ (-L— - — L \ _ J _ =
X„+2

\Xn + b

Xn+l+b)

°(^n+l - X„)

Xn + 2

(x„ + 6) (x„+i + 6)

=>(x„ + b){xn+x +b)= a{Xn + 2)(x„+i - x„)
=*-a;„x„+i + bx„ + bxn+i + 6^ = a ( x „ x „ + i - xl + 2x„+i - 2x„)
Sos
' ' ^ l i (1) va (2), thay rang can phai chon

1

1

1

x„ + 2

x„ - 3

x„+i - 3

,Vn€
a -

2a- 6

1
' 2:2+2

x„ +

a
1
2a - 6 5
1
b^

1 = 0,

2a+ 6

1

^

9

5'2a-6~5

2;
381

380

2

Li/u y. De t i n i bieu thiic sai phan —^-— = —^^
,,Vn € N * , sau
x„ + 2
x„ — 3 x„+i — 3
day ta se t r i n h bay mot phitdng phap kliac so \'di phudng phap t i m diem bat
dong ciia ham dac t n m g (da dudc t r i n h bay d l u u y ngay sau bai toan 260
CI trang 379), do la dua them tham so vao roi lira chpn. Gia siY

=^(2a - 6)x„+i + (a - l)x,.x„+i = ax^ + (2a + 6)x„ + 6^.

Do do

/

= Win (-

+ 2

(1)

G i a i . T£t (1) ta c6 x „ + i - x „ = ^ (x„ - 3)^ > 0, Vn = 1,2,... V?iy day so (xn)
5

Vay

n - + o c j-JXi

1
6 = -3.

/ a=

1

(2)

NhU v§,y, t a can cluing m i n h : — \

=

t o a n 263 (D^ t h i HSG t i n h Gia Lai-2003). Cho day s6 (u„) nhu sau:

^ — - , Vn e N * . Qua dav

ta thay rang phUdng phap s\ dung diem b i t dong ciia ham dac t r u n g gi^p
t a t i m r a Icli giai nhanh hdn so vcli phUdng phap dua them t h a m so vao rCi
lua chon.

" 2003

B a i t o a n 2 6 2 (De t h i HSG t i n h Gia L a i , nam hoc 2006-2007), Cho day
{xn}nTi

Vim

\U2
1—+00

^o.c dinh nhu sau: xi = \
=xl+3xn

Xn+l

(1)

l,n=l,2,3,...

+

U3

+ ••• +

G i a i . R u t g 9 n 5 „ = ^U2 + U3
g + ... + - ^ . D § t h a y t . „ > l , V n =

1,2,...

(*) ta CO
£'a<2/„ = E — ^ , V n = 1 , 2 , . . . rim

hm y „ .

j=l Xj + 2

ul = 2003(u„+i - u „ ) .

n-.+oo

Do do

G i a i . T i r (1) t a c6:

_ 2003K+1 - u „ )

Un

x„+i + 1 =

UnUn+l ;

+ 3x„ + 2 = (a;„ + 1) (x„ + 2), Vn = 1 , 2 , . . .

/
= 2003

Do do
1
^
1
^ _Jx„+,+1
( x „ + l ) ( x „ + 2)
x„ + l

1 _ Vn = 1 2
x „ + 2' "
'

"3

U2j

\Ux

/ I
= 2003 ( —
' i

Vxi+1

1
xi + 1

\

1

X2 + i y

1

\I

\X2 + l

1

1

x„+i + 1

2

-

^
Xn +

X3+1,

1

Xn+1 +

1/

hm

^

n->+oo X„+i

hm
1-.+00

3/„ =

lim
n-<+cxi

,Vn = 1,2,...

\

+ ••• +
1

= 2003

\

/ 1

1

, V n = 1,2,...

Tit gia thiet ta c6
(**)

/ I
2

-a day f . j ' , 6 gidi h,„ h«u ban. DS.
') cho n

x„+i > 3 " x i = 3 " , V n = l , 2 , 3 , . . .

n-.+co

\

I

quy nap t a de dang chi'mg uiinh ditdc

i i m x„+i = -l-oo

1

-

U-iJ

\U2

1
x„+i + r

De thay x„+i = xl + 3x„ + 1 > 3 x „ , Vn = 1 , 2 , 3 , . . . D o do bang phiTdng phap

D o do

1

U„+i

= 2003

I

/ I

Vay;
"2

/

= 2003

\UnUn+\i /

+ 0 0 t a difflc
a—

= 0. Bcii vay

2003

T i , (..) suy ra a > 1. T S

+ a <^ a = 0.

+ 1

1
\
x„+i + i ;

2

CO ^

(""^ '^'^^"S bi chan tren. Theo chu y 12 d trang 147

• ^ ^"
iim

Un = + 0 0 =j> lira

n—+oo

382

„„ =

n—>+oo

n-.+oo

383

= 0.
u„+i

uhi toan 265. Cho so thycc a > 1. Xit day so {x„} xdc dinh bdi xi = a vd.
^^^j==x„(l + x 2 0 i o ) , n > l . nm

Do (16:

lim

Sn

n—+oo

/

2003

= lim

1

= 2003 lim ( 1

1

n-'+oo

n-' + oo

\

= 2003.

/

)

^2010

.2010

Bai toan 264 (Olympic toan Siuh vieu loan qu6c-2010). Cho day so {^^j

n—+00

^ 2

.2010

X

lim

X

V/S? +

+ ^

y/Xn+l

+

xdc dinh bdi
Giai. Ttt gia thiet suy ra vdi mpi n ta c6 x„ > 1 va x^"" = x„+i - x„. Do do

X i = l , Xn+X = X „ ( 1 + 4 " " ' ) , 7 1 > 1 .
/_2010

Tim

_2010

_2010\'

"^i^H-

lim ( £ j _ + £ 2 _ ^ . . . ^ £ r ^ y

Giai. Tir gia thiet suy ra vdi iiipi n ta c 6 x„ > 1 va
„2oio

T

'2011

^2011

^2010

, _ X

1

Xn+l

a;„y'x„+i + x „ + i v ^

= Xn+\ x,,. Do (I5

^

y / ^ ^ -

^x„x„+i

_ _1

- Xn

+ v'x;^^)'

1

1

Vay
Vav
..2010

,.2010

j-2

X3

1
^Jl

.2010

^2010

X

'2010

,

X2

,

^2010
X3

^n+l
s/^n+l

1 \ '' 1

1

1

J-2y

a-3/

\Xn

+

1

£n+\
1

T\t gia thiet ta c 6

v/^

1 = J'l < X-2 < X3 < • • • < x„ < x„+i < • • •

_

1

^ J _ _

v/2;„+i

v/a

1

\/a;„+i'

(*)
Tir gia thiet ta c 6

\'ay (x„) la ciaj- tfiiig.

a = X i < X2 < X3 < • • • < X „ < Xn+l

Trifdng help 1: Day (x„) bi chan tren. Khi do theo djnh ly Weierstrass suy
ra day {x„) 16 gi(3i haii hitu han. Dat lim x„ = a. TCr (•) suy ra a > 1. Tif
= ^r. ( 1 + ^ ' f " ) • " ^ 1 f l ' " "
a = a + a^«"

+

^

o2""

'''^'^<-'

= 0 <^

1

0.

a =

Dieu nay man thuSii v6i a > 1 . Vay tritdug hO]) 1 khong tlie xay ra.
Trifdng hdp 2: Day so (x„) khong bi chan tren. Theo chu y 12 6 trang
ta CO
^
x„ = +cxi =>

lim
n-'+oo

lim
n-'+oo

= 0.
X„+i

lim

i—
X2

_20io

+ ^
X3

a;20io\

+ ...+,

^n+l

384

J

% (a;„) la day tang.
T^^Ung hdp 1: Day (x„) bi chan tren. Khi do theo dinh ly Weierstrass suy
^^ ^ay (a;„) ^6 gidi han hrni han. Dat lim x„ = a. Tit (*) suy ra a > 1. Tir
n—<+oo

*n+i = x„ ( 1 + x f i " ) , n > 1 cho n - » + 0 0 ta ditdc

1'

a = a + a^O"

a^"'! = Q ^ a = 0.

lim x„ = +00
=

= lim
n—+00

\

1.

"

P^eu nay man thuan vdi a > 1. V§,y trudng hdp 1 khong the xay ra.
«c»ng hdp 2: Day so (x„) khong bj chan tren. Theo chii y 12 d trang 147
Co

Do do
.2010

< •••

n-»+oo

lim ,
n - » + o o ^Xn+\

)

385

= 0.

V, sn+i

Do (16
X.2010

lim

•+

X2

n-'+oo

-2010

„2010
^2

• v/^^n+l
+

nen

x„+i - 1 > 1 ^

0 <

+

no do [52013] = 0 va

[5„]=0,Vn=l,2,...

^
Xn+l

-

1

lim 5 „ = 1.

Q'ai toan 267 (De nghi OLYMPIC 30/04/2004). TH day so (u„) du^Jc xdc
dinh bdi:

1
=

> 2

lim

ui=2

Un+l

Bai toan 266. Xet day s6 { x , . } ^ ! n/iU sau: x, = 2va
(1)

=

ul + 2003u„
2004

ta thdnh lap day so (Sn) vdi Sn=Yl

Tim

lim 5„.
n—•+00

Ddt

+ • • • + T-r—

+ z-^—

Sn = r - ^ ^ —

Hdy tim phan nguyen cm S2013 va

^

tinh gidi han cua 5„ khi n tang len vd han.
Giai. Ta chi'mg luinh day so

x„+i - x-„ - i ( 4 + 1) - x„ = ^ (a;„ - 1)2 > 0, Vn = 1,2,...
Vay {x„}+f°i la day so tang va x,, > xi = 2, Vn = 1,2,... Mat khac iieu
{xn}t=i

14 flian

tren tin day nay se c6 gidi han hQu han. Gia sii

Xn =

a,

khi d6 a > 2. Tir (1) cho n -+ +00 ta dUdc
a-^{a'^ + 1)

•^a=l.

Den day ta gap di§u niau thnan v6i a > 2. Vay day so {xn}n=i
tren. Suy ra lim x„ - +00. V6i inpi n = 1,2,..., ta c6

khong bi clia"

-

1 = i

+ 1) -

1

1

1 = i (x„

1

Ta rhi'mg ininh ditflc

-

1) ( X „ +

1

.

1)

Ul - 2004[;„+, + 2003C/„ = 0 (n = 1,2,...).

1
^"'taJe' lim

Do (16
5„-

1
Xl -

1

X2 -

A

1

+

/
\X2

= 1
Xl - 1

x„+i - 1

1

1
-

1
1

lim S„ = 2004.

Bai toan 268 (Dl nghi thi OLYMPIC 30/04/2004). Cho day s6 ((/„) xdc
'^mh hdi. (/, = a (a > 0 c/i(^
n—+oo
Xn+l

uj + 2003un ^ K - l)(u„ + 2004)
^r;:^.
.1
2004
2004
2004
2004 / 1
1
Un+l - 1
{Un - l)(u„ + 2004)
2005 \ U n - \ 2004^
2005
2004
2004
2004
2004
2005
Un+l - 1
- 1 Un + 2004
Un + 2004
Un - 1
U„+i - 1
2004
1 _ 2005
2005
u„ + 2004 Un - 1 u„ - 1 M „ + l - 1
2005 u„
2005
2005
ul + 2003u„ - 2004 u„ - 1 u „ + i - 1
2004
1
1
2004
2004u„+i - 2004 Un - 1 " n + l - 1 " " n + l " ^
~^
^

Un+i - ^ =

ta"g va khoiig bi cli^Lii tren. Ta c6

{xn}n=i

Hi^<3ng dan. Tir gia thiet ta c6

X3 -

^
1

+ ••• +

/

1
X„ - 1

Xn+l

• I,

£77-^=2004.

d a n . Ta c6
hm

Xn+l -

v-^

>
1=1

Ui

=

1

386

387

;^uu4
2004

a= 2

Cho day so ( x „ )

B a i toan 269.

( n = 1 , 2 , . . . ) diKic xdc dinh nhu

sau:
h d p 1: D a y { / ( n ) } + f = i

r^xidng
X I = 1; x „ + i = v / x „ ( x „ + l ) ( x „ + 2 ) { x „ + 3) + 1 , V n =

1,2,...

b i chan t r e n . K h i do v i day { / ( n ) } + ~

b i c h a n t r e n n o n c6 g i d i h a n h f l t i h a n . D a t

lim

0 < a = / ( I ) < / ( 2 ) < •.. < / ( n ) < / ( n + 1) < . . .
guy r a I > 0. Tit f{n
H i f d n g dSn. T i i Xn+i =

V [ x „ ( x „ + 3)] [(x„ + l ) { x „ + 2)] + 1 , V n =

+ 1) = 2 0 0 1 / 2 ( n ) + / ( n ) cho n ^

1,2,...

"

+oo t a d i w c

'
fi;: r

L = 20011,2 + L -izt' L = 0.

t a CO
x „ + i = v / ( x 2 + 3 x „ ) ( x 2 + 3 x „ + 2) + l

tang

/ ( n ) = L . Tit

D i 6 i i n a y m a n t h u a n v d i L > 0. V a y t r U d n g h d p 1 k h o n g t h e x a y r a .

,Vn=l,2,...

I V U d n g h d p 2: D a y so { / ( n ) } + ~ k h o n g b i c h a n t r e n . K h i d o
y{x2+3x„)2 + 2(x2+3x„) + l

=

= ^[(x2+3x„) + lp

,Vn=l,2,...

=x2+3x„ +

l i m / ( n ) = +00 => h m / ( n + 1) = + o o =^ l i m - — ^ — - = 0.
n-+oo
n-.+oo
„^+oo/(n+l)

l,Vn=l,2,...

Do do

Sau d o s i i d u n g b a i t o a n 2 6 2 a t r a n g 382.
B a i toan 270

( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Day

so ( x „ )

dMc

nhu sau: x i = 1 vd
xn+i

= x / x „ ( x „ + 5) ( x 2 + 5 x „ + 8) + 16, V n =

1
,
. Tim
i=l Xi + 3

Bai t o a n 272.

l,2,...

xdc dmh

tren

[1; +oo)

vd thoa man

/(I)

Cho

Tim

1
lim
n-.+oo \ + X i

l)J

+

lid

2001a'

xi = a > 0 vd

lim
n-.+oo

N

' 1 + X„
1

T

Xi - 1

, biet r&nq
^

x i = 3; x „ + i = x^ - 3 x „ + 4, V n e N * .

tinh
lim
n^+oo

fin)
[/(2)

"

G i a i . T i t gia t h i e t suy r a / ( n )

/(3)
>

/(I)

B a i t o a n 2 7 4 ( C h i n a G i r l s M a t h O l y m p i a d - 2 0 0 3 ) . Cho day so (a„)+f°i
sau: a i = 2 w

f{n+l)\
=

a >

0, V n =

1,2,...

_

2001/^(n)

^

2001/(n)/(n+l)

D e t h a y d a y so {f{n)}'^^^

/(n+l)-/(n)
2 0 0 1 / ( n ) / ( n + 1)

^ J _

Chiing minh
[JL_

_ _ _ 1

2001 [ / ( n )

/(n + D

nhu

a„+i = a ^ - a „ + l , V n € N * .

V d i niQi

n = 1, 2 , . . . , t a CO / ( n + 1) = 2 0 0 1 / 2 ( n ) + / ( n ) . S u y r a

/ ( n + 1)

1

+

Tim

/ ( x + 1) = 2 0 0 1 / 2 ( x ) + / ( x ) , V x e [ 1 ; + o o ) .

/(n)

1
1 + X2 '

a > 0 ud
Bai t o a n 273.

Hay

day so ( x „ ) n / i t / sau:

Cho

1
f{n

x„+i = x ^ + x „ , V n = 1,2,...

lira y „ .
n-.+oo

B a i t o a n 271 (De t h i O L Y M P I C t o a n Sinh V i e n t o a n quoc n a m 2001).
ham so f{x)

1
hm
2001 u - + o o V a

fjk)
lim
V
n—+oo ^ / ( f c + 1)

xdc dmh

1

rdng 1
^

^
<
20032003

(1)

2003 1
E - < i i=i at

' a i - T a CO a „ + i - a „ = ( a „ - 1)^ > 0, V n 6 N * . Tilt d a y v a do a i = 2 n e n

l a d a y so dUdng v a d d n d i e u t a n g . T a c6

2 =ai

< a2 < ••• < an < a„+i

(2)

<

^^tkha<;
V/(fe)
2 - / ( A ; + 1) "

_ L V
[ J
2001 £ ^ [ / ( f c )

^1
= J - ( - l
/(A:+1)J
2001 V / ( I )

^—
fin+D

l_
"i+i

— 1

'+1 - 1

a „ ( a „ - 1)

a„+i - 1

a„ - 1

388
389

a„

a„

a„ - 1

a„+i

-

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