Let n be a positive integer.
(a) If (1+px+x^2)^n = 1+ax+bx^2+cx^3+terms involving higher powers of x,
express a,b and c in terms of p and n.
(b) If the coefficients of x and x^2 in the expansion of (1+px+x^2)^n are 14 and 91 respectively, find the values of p and n.
(c) Hence find the coefficient of x^3.
Binomial
2010-04-18 8:02 pm
回答 (1)
2010-04-18 8:25 pm
✔ 最佳答案
(1+px+x²)ⁿ=1+nx(p+x)+(1/2)(n)(n−1)x²(p²+2px+...)+(1/6)(n)(n−1)(n−2)x³(p³+...)+.....
=1 + npx + [n+(1/2)(n)(n−1)p²]x² + [n(n−1)p+(1/6)(n)(n−1)(n−2)p³]x³ +.....
(a)
a=np
b=n+(1/2)(n)(n−1)p²
c=n(n−1)p+(1/6)(n)(n−1)(n−2)p³
b)
np=14
p = 14/n _____(1)
n+(1/2)(n)(n−1)p²=91 ______(2)
Put (1) into (2):
n+(1/2)(n)(n−1)(196/n²)=91
n²+98(n−1)=91n
(n−7)(n+14)=0
n=7 or n=−14(rej.)
∴p=2
c)
Coefficient of x³ = n(n−1)p+(1/6)(n)(n−1)(n−2)p³ = 364
收錄日期: 2021-04-23 18:23:59
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