HD Giai Cac Bai Toan Ve Nhi Thuc Niwwton

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TL rn luyn thi H-C - NBS Phm Huy Hot 11 - 2012----------------------------------------------------------------------------------------------------------------------

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HNG DN GII CC BI TON V NH THC NEWTON

Gii thiu: Cc bi ton v Nh thc Newton gn y rt hay gp trong cc thi khi A (H-C) ; ra khng kh, ch cn nm vng cng thc/nh l l gii

c. V HS thng it tip cn vi dng ny nn lng tng gii mt nhiu thigian. Ti liu ny gip cc bn h thng li phn kin thc lin quan v su tm mtlot ton thi H c gii bng ng dng Nh thc Newton cc bn tham kho. Ni dung chnh trong ti liu l ca bn Nguyn Trung Hiu, NBS ch sp xp lai, cccng thc, cc k hiu ton hc u bin son bng latex- Tng phn, tng bi tonc t trong khung rt tin cho ngi s dng khi cn sao trch, bin son bi gingcho HS.

A.- Phn L THUYT cn nm vng:

1/.Cc hng ng thc lin quan

0

1

2 2 2

3 3 2 2 3

4 4 3 2 2 3 4

1

2

3 3

4 6 4

...

a b

a b a b

a b a ab b

a b a a b ab b

a b a a b a b ab b

2.-Nh thc Newton( Niu-tn)

a/.nh l:

0 1 1 1 10

...n

n n n n n n n k n k k

n n n n n

k

a b C a C a b C ab C b C a b

H qu:

* 0 0 1

kn nnn kk n k k n k k

n nk ka b a b C a b C a b

* 0 1

0

1 . . ... .n

n k k n n

n n n n

k

x C x C C x C x

b/.Tnh cht ca cng thc nh thc Niu-tn n

a b :


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-S cc s hng ca cng thc n

a b l n+1-Tng s m ca a v b trong mi s hng lun lun bng s m ca nh thc:

(n-k)+k=n

-S hng tng qut ca nh thc l: 1k n k k

k nT C a b

( l s hng th k+1 trong khai trin n

a b )

-Cc h s nh thc cch u hai s hng u, cui th bng nhau.1 02 ...n n nn n nC C C

0 10 ... 1n n

n n nC C C

-Tam gic pascal:Khi vit cc h s ln lt vi n = 0,1,2,... ta c bng

n k

0 1 2 3 4 5 ....

0 1

1 1 1

2 1 2 1

3 1 3 1

4 1 4 6 4 1

5 1 5 10 10 5 1

Trong tam gic s ny, bt u t hng th hai, mi s hng th n t ct thhai n ct n-1 bng tng hai s ng hng trn cng ct v ct trc n.

S d c quan h ny l do c cng thc truy hi1

1 1

k k k

n n nC C C (Vi 1 < k< n)

3/.Mt s cng thc khai trin hay s dng:


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3

1 00

2 1 1 ...n

nn k n n

n n n n

k

C C C C

0 10

0 1 1 1 ... 1n

n k nk n

n n n n

k

C C C C

0 1 1 00

1 ...n

n k n k n n n

n n n n

k

x C x C x C x C x

0 0 1 10

1 1 ... 1n

n n nk k n n

n n n n

k

C x C x C x C x

0 1 1 00

1 1 ... 1n

n k nk n k n n n

n n n n

k

x C x C x C x C x

4/.Du hiu nhn bit s dng nh thc newton.

a/.Khi cn chng minh ng thc hay bt ng thc m c1

ni

n

i

C

vi i l s t

nhin lin tip.

b. Trong biu thc c 1

1n

i

n

i

i i C

th ta dng o hm i

Trong biu thc c 1

n

in

i

i k C

th ta nhn 2 v vi xkri ly o hm

Trong biu thc c1

nk i

n

i

a C

th ta chn gi tr ca x=a thch hp.

Trong biu thc c1

1

1

ni

n

i

Ci

th ta ly tch phn xc nh trn ;a b thchhp.

Nu bi ton cho khai trin

1 1

i

n nn n i a n i iba b i a b i

n n

i i

x x C x x C x

th h s ca xm l Cin sap

cho phng trnh a n i bi m c nghim i

inC t MAX khi 12

ni

hay

1

2

ni

vi n l,

2

ni vi n chn.


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B.- CC BI TON NG DNG NH THC NEWTON.

I.-Cc bi ton v h s nh thc.

1/.Bi ton tm h s trong khai trin newton.

Bi ton 1: ( thi H Thu li c s II, 2000)Khai trin v rt gn a thc:

9 10 14

1 1 ... 1Q x x x x

Ta c a thc: 14

0 1 14...Q x a a x a x Xc nh h s a9.

Gii:

H s x9 trong cc a thc 9 10 14

1 , 1 ,..., 1x x x ln lt l:9 5 9

9 10 14, ,...,C C C

Do :

9 5 9

9 9 10 14

1 1 1 1... 1 10 .10.11 .10.11.12 .10.11.12.13 .10.11.12.13.14

2 6 24 20a C C C

a9 =11+55+220+715+2002=3003

Bi ton 2:(HBKHN-2000)Gii bt phng trnh: 2 2 32

1 610

2x x xA A C

x

Gii:iu kin: x l s nguyn dng v 3x Ta c: dt phng trnh cho tng ng vi:

2 1 2 6 2 11 10

2 3!

2 2 1 2 2 1 103 12 4

x x x xx x

x

x x x x x xx x

V x l nghim nguyn dng v 3x nn 3;4x

Bi ton 3: (H KA 2004)

Tm h s ca x8 trong khai trin a thc ca: 82

1 1x x


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Gii:

Cch 1: Ta c: 8 8

2 2

8 8

0 0 0

1 1 .

kk kik k k i i

k

k k i

f x C x x C x C x

Vy ta c h s ca x8 l: 81i k i

kC C tha mn

00 8

42 8

2,

3

ii k

kk i

ii k

k

H s trong khai trin ca x8 l: 0 24 0 3 2

8 4 8 31 1C C C C =238

Cch 2: Ta c:

3 4 8

0 3 2 4 2 8 2

8 8 8 8... 1 1 ... 1f x C C x x C x x C x x

Nhn thy: x8 ch c trong cc s hng:

S hng th 4: 33 28 1C x x S hng th 5: 44 28 1C x x Vi h s tng ng vi: A8=

3 2 4 0

8 3 8 4C C C C =238

Bi ton 4:(H HCQG, 2000)

a) Tm h s x8 trong khai trin 1211x

b) Cho bit tng tt c cc h s ca khai trin nh thc 2 1 nx bng1024. Hy tm h s a *a ca s hng ax12 trong khai trin .

( HSPHN, khi D,2000)

Gii:a) S hng th (k+1) trong khai trin l:

12 12 2

12 12

1k

k x k k

ka C x C x

x

0 12k

Ta chn 12 2 8 2k k

Vy s hng th 3 trong khai trin cha x8 v c h s l:2

12 66C

b) Ta c: 2 2 1 2 12 20

1 ...n

k n k k k

n n n n

k

x C x C C x C x

Vi x=1 th: 0 12 ... 1024n nn n nC C C

102 2 10n n


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Do h s a (ca x12) l:6

10 210C

Bi ton 5:(HVKTQS, 2000) Khai trin a thc:

12 120 1 12(1 2 ) ...P x x a a x a x

Tm max 0 1 2 12, , ,...,a a a a

Gii:Gi ak l h s ln nht ca khai trin suy ra: 1k ka a T y ta c h phng trnh:

1 1

12 12

1 112 12

2 1

2 2 12 1

1 22 212 1

k k k k

k k k k

C C k k

C Ck k

8 180 1 2 12 8 12ax , , ,..., 2 126720m a a a a a C

2/.Bi ton tm s hng trong khai trin newton.

Bi ton 6: Tm s hng th 21 trong khai trin: 25

2 3x

Gii:S hng th 21 trong khai trin l:

2020 5 20 5 20 20

25 252 3 2 3C x C x

Bi ton 7:a. Tm s hng ng gia trong cc khai trin sau 213x xy b. Tm s hng ng gia trong cc khai trin sau

20

4

23

1x x

xy

Gii:

a. Khai trin 203x xy c 21+1=22 s hng nn c hai s hng ng gia l sth 11 v 12.

S hng th 11 l: 11 1010 3 10 43 1021 21C x xy C x y S hng th 12 l: 10 1111 3 10 41 1121 21C x xy C x y


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b. Khai trin

20

4

23

1x x

xy

c 20+1=21 s hng. Nn s hng ng gia 2 s l

s hng th

10 10 65 207 210 10 6 34

320 20

21

1 16 :2 C x xy C x y

( Vi [x] l k hiu phn nguyn ca x ngha l s nguyn ln nht khng vt qu x).

Bi ton 8: (H Khi D-2004) Tm s hng khng cha x trong khai trin.

7

3

4

1f x x

x

vi 0x

Gii:

S hng tng qut trong khai trin:

7 77

3 3 12

1 7 74

1

, 7

kk k

k k

kT C x C x k k x

ng vi s hng khng cha x ta c: 7 7 0 43 12

k k

Vy s hng khng cha x trong khai trin f x l: 47 35C

Bi ton 9: (H SPHN-2001) Cho khai trin nh thc:10

9 10

0 1 9 10

1 2... .

3 3x a a x a x a x

Hy tm s hngk

a ln nht.

Gii:

Ta c: 10

10

10 1010 10 100

1 2 1 1 11 2 2 2

3 3 3 3 3

nkk k k

k

k

x x C x a C

Ta c akt c max

1 1

1 10 10

1 11 10 10

2 2

2 2

2 10! 2 10! 1 2! 10 ! 1 ! 9 ! 19 2210 1

2 2 3 32 10! 2 10!

11! 10 ! 1 ! 11 !

7 , 0,10

k k k k

k k

k k k k k k

k k

k k

a a C C

a a C C

k k k k k kk

k kk k k k

k k k

Vy max7

7

7 1010

2

3k

a a C


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Bi tp p dng

Bi 1: (H TK-2002) Gi a1, a2,, a11 l cc h s trong khai trin sau: 11 101 111 2 ...x x x a x a

Hy tm h s a5

Bi 2: ( Khi D-2007)Tm h s ca x5 trong khai trin

5 1021 2 1 3x x x x

Bi 3:( 4 TH&TT -2003)Tm h s ca x5y3z6t6 trong khai trin a thc

20x y z t

Bi 4: (TT H- chuyn Phan Bi Chu-Ngh An) Xc nh h s ca x11 trongkhai trin a thc: 2 32 3 1

n n

x x bit:

2 2 1 2 2 02 2 2 23 ... 1 3 ... 3 1024kn n k n k n

n n n nC C C C

Bi 5: (LAISAC) Khai trin 3 21

2

n

P x xx

ta c

3 3 5 3 100 1 2 ...n n nP x a x a x a x

Bit rng ba h s u a0, a1, a2 lp thnh cp s cng. Tnh s hng th x

4

II. p dng nh thc Newton chng minh h thc v tnh tng t hp.

1/.Thun nh thc Newton

Du hiu nhn bit: Khi cc s hng ca tng c dngk n k k

nC a b

th ta s dng

trc tip nh thc Newton: 0

nn k n k k

n

k

a b C a b

.Vic cn li ch l kho lo chn a,b.

Bi ton 10: Tnh tng 16 0 15 1 14 2 1616 16 16 163 3 3 ...C C C C

HD Gii:D dng thy tng trn c dng nh du hiu nu trn. Ta s chn a=3, b=-1.Khi tng trn s bng (3-1)16 = 216

Bi ton 11: ( H Hng Hi-2000) Chng minh rng:


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0 2 2 4 4 2 2 2 1 22 2 2 23 3 ... 3 2 2 1n n n nn n n nC C C C Gii:

2 0 1 2 2 2 1 2 1 2 2

2 2 2 2 2

2 0 1 2 2 2 1 2 1 2 2

2 2 2 2 2

1 ... 1

1 ... 2

n n n n n

n n n n n

n n n n n

n n n n n

x C C x C x C x C x

x C C x C x C x C x

Ly (1) + (2) ta c:

2 2 0 2 2 2 2

2 2 21 1 2 ...n n n n

n n nx x C C x C x

2.S dng o hm cp 1,2.

a/. Dng o hm cp 1.

Du hiu: Khi h s ng trc t hp tng dn hoc gim dn t 1,2,3,,n

hay n,,3,2,1 tc l s hng c dngk

nkC hoc

1k n k k

nkC a b th ta c th

dng o hm cp 1 tnh. C th:

0 1 12 ...n n n n n

n n na x C a C a x nC ax

Ly o hm hai v theo x ta c:

1 1 1 2 2 12 ... 1

n n n n n

n n nn a x C a C a nC ax

n y thay x,a bng hng s thch hp ta c tng cn tm.

Bi ton 12:(H BKHN-1999) Tnh tng

11 2 3 42 3 4 ... 1

n n

n n n n nC C C C nC

2 2 0 2 2 2 2

2 2 2

4 20 2 2 2 2

2 2 2

2 20 2 2 2 2

2 2 2

2 1 2 0 2 2 2 2

2 2 2

4 2 2 3 ... 3

2 23 ... 3

2

2 2 13 ... 3

2

2 (2 1) 3 ... 3

PCM

n n n n

n n n

n nn n

n n n

n nn n

n n n

n n n n

n n n

C C C

C C C

C C C

C C C

Chn x=3 suy ra:


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Gii:Ta thy tng cn tnh c dng nh cng thc (1). Vic cn li ch cn chn a=1,x=-1ta tnh c tng bng 0.

Cch khc: S dng ng thc 11k kn nkC nC ta tnh c tng bng:

1 10 1 2 1

1 1 1 1... 1 1 1 0

n nn

n n n nnC nC nC nC n

Bi ton 13:Tnh tng:0 1 2007

2007 2007 20072008 2007 ...C C C

HD Gii:H s trc t hp gim dn t 2008,2007,,1 nn dng o hm l iu d hiu:

2007 0 2007 1 2006 2007

2007 2007 20071 ...x C x C x C By gi nu o ly o hm th ch c 0 200620072007C x trong khi n 2008 do ta phi nhn thm vi x vo ng thc trn ri mi dng o hm:

2007 0 2008 1 2007 2007

2007 2007 2007

2006 0 2007 1 2006 2007

2007 2007 2007

1 ...

1 2008 1 2008 2007 ...

x x C x C x C x

x x C x C x C

Thay x=1 vo ta tm c tng l 2009.22006

b/. Dng o hm cp 2.

Du hiu: Khi h s ng trc t hp c dng 1.2,2.3,,(n-1)n hay(n-1)n,,3.2,2.1 hay 1

2,2

2,,n

2(khng k du) tc c dng ( 1) k n k

nk k C a

hay tng qut hn 1 k n k k nk k C a b th ta c th dng o hm n cp 2

tnh. Xt a thc 0 1 1 ...

n n n n n

n n na bx C C a bx C b x

Khi o hm hai v theo x ta c:

1 1 1 2 2 2 12 ...n n n n n n

n n nbn a bx C a b C a b x nC b x

o hm ln na: 2 2 2 2 2 11 2.1 ... 1 2n n n n nn nb n n a bx C a b n n C b x

n y ta gn nh gii quyt xong v d ton ch vic thay a,b,x bi cchng s thch hp na thi.


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Bi ton 14: (H AN-CS Khi A 1998)

Cho 1 , 2n

f x x n

a.Tnh 1f

b.Chng minh rng:

2 3 22.1 3.2 ... 1 1 2n nn n nC C n nC n n

Gii:

a. 1 2 21 1 1 (1) (1 )

n n nf x n x f x n n x f n x

b. Ta c

0 1

1 2

1 1

2

2

2

2

1

1 2 2 1

1

1

1 1 2

2.1 3.2 ... 1 ... 1 1 2 PCM

n nn k k k k

n n n n

k k

nk k

n n

k

nk k

n

k

nk n

n

k

p n nn n n n

f x x C x C C x C x

f x C kC x

f x k k C x

f k k C

C C p C n nC n n

T cu b thay (n-1)=(n+1) th ta c mt bi ton khc:b. Chng minh rng:

1 2 22.1 3.2 ... 1 ... 1 1 2p n nn n n nC C n pC n nC n n

Vi bi ton ny ta gii nh sau:

Xt nh thc: 0 11 ...

n n n

n n nx C C x C x

Nhn 2 v ca ng thc vi 0x ng thi ly o hm cp 2 hai v theo bin x tac

: 1 2 1 2 12 1 1 1 2 3.2 ... 1

n n n n

n n nn x n n x x C x C x n nC x

Cho x=2 ta c PCM


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Bi tp p dngBi 1:(CSP Bn Tre Khi A-2002) Chng minh rng:

1 1 19 19

20 20 20... 2C C C

Bi 2:(C Khi T-M-2004)Chng minh rng :

20040 2 1 2004 2004

2004 2004 2004

3 12 ... 2

2C C C

Bi 3:(HKTQD-2000) Chng minh:

1 1 2 2 2 2 12 1.2 . 2.2 . 3.2 . ... .3 1n n n n n n

n n n n

x C C C nC n n

Bi 4: Rt gn tng:

2 1 2008 2 2 2007 2 2009

2009 2009 20091 2 2 2 ... 2009C C C

III.Mt s phng php khc:

Bi ton 15: (HQG TP.HCM 1997) Cho

0

, ,

m k n

k m n Z

Chng minh:0 1 1. ...k k k m m k n m n m n m n mC C C C C C C

Gii:

0 1

0 1 1

0 1

1 ...

Ta c : 1 ...

1 ...

m m m

m m m

n n n n

n n n

m n m n m n

m n m n m n

x C C x C x

x C x C x C

x C C x C x

Suy ra h s xk trong (1+x)n .(1+x)m l 0 1 1 ...k k m k mm n m n m nC C C C C C

V h s xk trong khai (1+x)m+n lk

m nC

ng nht thc: (1+x)n .(1+x)m = (1+x)n+m


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Ta c:0 1 1. ...k k k m m k n m n m n m n mC C C C C C C

PCM

Bi ton 16: (2-TH&TT-2008)

S2=

2 2 21 22 ... n

n n n

C C n C

vi n l s t nhin l

Gii:Ta c:

2 2

1 12 2 2

1 1 2 21 1

1 ...2 2

n n

n n

n n n n n

n nS C n C C C n C

2 2 21 2 1

2 2 21 2 1

2 2 21 2

...

...

2 ...

n

n n n

n n

n n n

n

n n n n

n C C C n

n C C C n

S n C C C n

Mt khc ta c: 2 0 1 2 2

2 2 21 ...

n n n

n n nx C C x C x h s ca xn

l: 2 (*)n

nC

Trong khi : 0 11 ...n n n

n n nx C C x C x

Nn h s ca xn l 2 2 2

1 2 ... nn n n

C C C (**)

T (*) v (**) 2 2 21 2

2 1 ...n n

n n n nC n C C C

2 PCM2

n

n n

nS C

Bi tp p dngBi 1: Chng minh rng:

a)1 1 2 1 13 2 3 ... .4

n n n n

n n nC C nC n (H Lut-2001)

b) 2 1 2 2 2 21 2 ... 1 2n nn n nC C n C n n ( 1-TH&TT-2008)

Bi 2: Tnh cc tng sau:

a)1 2 3 4 5 28 29

30 30 30 303.2 5.2 ... 29.2C C C C

b) 1 2

0 ... 12 3 1

nnn n n

n

C C CC

n

Bi 3: t 1 2 1

61 3

k k k

k nT C

. Chng minh3

1

0n

k

k

T


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