rut gon tong tich huu han.pdf

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  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 1

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 2

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 3

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 4

    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

    .

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 5

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 6

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 7

    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.

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 8

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 9

    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

  • V Tin Trnh Trng PTNK

    toan999.wordpress.com 10

    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