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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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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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