Rut Gon Tong Tich Huu Han

V Tin Trnh Trng PTNK

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BI TP RT GN CC TNG V TCH HU HN

A.Rt gn tng.

Mt s biu thc thng gp.

a)

1 1 11 1k k k k

b) 21 1 1

1k kk

c)

1 1 121 1k k k k

d)

1 1 1 12 12 1 1 k kk k k

e)

1 1 11 1 1k k k k k k

Chng minh.

a)

1 1 1 11 1 1

k kk k k k k k

b) 2

1 1 1 11 1k k k kk

c)

1 1 2 1 11 11 1 1 1

k k k k k k kk kk k k k k k k k

2 1 1 121. 1

k k

k k k k



d)

1 1 1 1 1 11 2 12 12 1 1

k k k kk k k kk kk k k

e)

2 2

1 1 1 1111 1 1 1

k k k k k k k kk kk k k k k k k k

1 11k k

V d 1. Cho s nguyn dng 2n . Tnh tng

1 1 1...1.2 2.3 1

Sn n

Gii.

p dng

1 1 11 1k k k k

Ta c : 1 111.2 2

1 1 12.3 2 3

.

1 1 11 1n n n n

Vy 11Sn

V d 2. Cho s nguyn dng n . Chng minh 2 2 2 21 1 1 1 2 1...1 2 3

nnn

Gii.

p dng 21 1 1

1k kk

vi 2k



Ta c : 21 11

22

21 1 1

2 33

21 1 1

1n nn

Do 2 2 21 1 1 1... 12 3 nn

Vy 2 2 21 1 1 1 2 1... 1 11 2

nn nn

V d 3. Cho s nguyn dng 2n .

Rt gn 1 1 1...1 2 2 3 1n

Sn n

Gii.

S dng lng lin hip ta c: 1 11

k kk k

Do 2 1 3 2 ... 1 1nS n n n

V d 4. Cho s nguyn dng n .

Chng minh

1 1 1 1 2... 22 3 2 4 3 1 1n n n

Gii.

p dng

1 1 121 1k k k k

vi 1k



Ta c: 1 12 12 2

1 1 123 2 2 3

1 1 124 3 3 4

..

1 1 121 1n n n n

Cng v theo v ta c

1 1 1 1 1 2... 2 1 22 3 2 4 3 1 1 1n n n n

V d 5. Vi n l s nguyn dng.

t

1 1 1...2 1 1 2 3 2 2 3 1 1n

Sn n n n

Chng minh 1nS

Gii.

p dng

1 1 11 1 1k k k k k k

Ta c: 1 112 1 1 2 2

1 1 13 2 2 3 2 3

.



1 1 11 1 1n n n n n n

Cng v theo v ta c 11 11n

Sn

V d 6. Cho n l s nguyn dng. Chng minh

1 1 12 3 1 ... 2 22 3

n nn

Gii.

Ta c: 1 2 2 2 12 1 k kk k k k

1 2 2 2 12 1 k kk k k k

Vy 12 1 2 1k k k kk

p dng cho 2,3,4,...,k n ta c

12 3 2 2 2 12

12 4 3 2 3 23

..

12 1 2 1n n n nn

Cng v theo v ta c

1 12 1 2 ... 2 12n nn



M 2 1 2 2 2 3n n nn ta c iu phi chng minh.

Bi tp.

Bi 1. ( tuyn sinh lp 10 chuyn Thi Bnh 2009)

Chng minh rng 1 1 1 1 88...2 453 2 4 3 2010 2009

Bi 2. ( tuyn sinh lp 10 chuyn Ton Bnh nh 2010).

Cho s t nhin , 3n n .

t

1 1 1...3 1 2 5 2 3 2 1 1

nSn n n

Chng minh 12n

S

Bi 3. ( tuyn sinh lp 10 chuyn Hng Vng tnh Ph Th 2001)

Chng minh rng

43 1 1 1 44...44 452 1 1 2 3 2 2 3 2002 2001 2001 2002

Bi 4. Chng minh

1 1 12014 1 ... 20152 3 1016064

Bi 5. Cho n l s nguyn dng. Chng minh

1 1 1 1... 22 3 2 4 3 1n n

Bi 6. Cho n l s nguyn dng. Tnh tng

2 2 2

3 5 2 1...1.2 2.3 1

nnS

n n



Gi : Chng minh v p dng 2 2 22 1 1 1

11k

k kk k

B. a v cng mu.

V d 7. Cho 2n l s nguyn dng.

t

1 1 1 1... ...1. .12. 1 1

Sn nn k n k

Hy so snh S vi 21

nn

Gii.

Vi ,a b hai s dng phn bit, ta c

2 1 20 2a b a b ab a bab

p dng ta c:

1 211. nn

1 2 2

2 1 12. 1 n nn

.

1 21.1 nn

Cng v theo v ta c 21n

nSn

Bi tp.



Bi1 . Cho

1 1 1 1... ...1.2016 2.2015 2016.12016 1

Sk k

Hy so snh S vi 201622017

Bi 2. Cho

1 1 1 1... ...1.1999 2.1998 1999.11999 1

Sk k

Hy so snh S vi 1,999

C. Rt gn tch.

V d 8. Cho 2n l s nguyn dng. Rt gn 1 1 11 1 .... 12 3n

Pn

Gii

p dng 1 11 kk k

vi 2,3,4,...,k n ta c :

3 4 5 1 1...2 3 4 2n

n nPn

V d 9.

Chng minh

1.3.5.... 2 11 12.4.6.... 22 2 1

nnn n

vi mi s nguyn dng n .

Gii.

Ta c : 2

2 2

2 12 1 2 1 2 12 2 14 4 1

kk k kk kk k

Ta c: 2 2 22 1 4 4 1 4 4 4 1k k k k k k k



2

2 22

2 1 12 1 4 14k kk k k k

kk

2 1 12k k

k k

Vy 1 2 1 2 12 2 1

k k kkk k

p dng cho 1,2,...,k n ta c:

1 12 3

1 3 342 5

2 5 563 7

.

1 2 1 2 12 2 1

n n nnn n

Nhn v theo v ta c

1.3.5.... 2 11 12.4.6.... 22 2 1

nnn n

Bi tp.

Bi 1. Cho 2n l s nguyn dng. Rt gn 1 1 11 1 ... 12 3n

Pn

Bi 2. Cho 2n l s nguyn dng. Rt gn 2 2 21 1 11 1 ... 12 3n

Pn



Gi : chng minh v p dng 2

111

1

kkkk

k

Bi 3. Cho 2n l s nguyn dng. Rt gn 3 3 3

3 3 32 1 3 1 1. ....2 1 3 1 1n

nPn

Gi : Chng minh v p dng

3

3

1 11 1. .1 1 11

k kk k kk k k kk

Bi 4. Cho n l s nguyn dng.

t 22 1 1na n n

Tnh tch 1 3 2 12 4 2

. ..... ....

n

n

a a aTa a a

Gi . Chng minh v p dng

22 1

22

2 1 12 1 1

k

k

kaa k

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Rut Gon Tong Tich Huu Han