F4 Binomial Expansion超急15點!!!!

Larry

2009-12-18 5:28 am

a) Expand(x+1)^n in ascending powers of x up to the 3rd term,where n is a
positive integer.
b) It is given that (x+1)^n(1-x+2x^2)=1+8x+kx^2+terms involving higher powers
of x.Find the values of n and k.

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更新1:

From the given:n - 1 = 8, <---邊度講n-1=8??????

回答 (1)

六呎將軍

2009-12-18 5:36 am

✔ 最佳答案

(a) (x + 1)n = nC0 + nC1 x + nC2 x2 + ...

= 1 + nx + n(n - 1)x2/2 + ...

(b) (x + 1)n(1 - x + 2x2) = [1 + nx + n(n - 1)x2/2 + ...] (1 - x + 2x2)

= [1 + nx + n(n - 1)x2/2 + ...] - x[1 + nx + n(n - 1)x2/2 + ...] + 2x2[1 + nx + n(n - 1)x2/2 + ...] + ...

= 1 + nx + n(n - 1)x2/2 - x - nx2 + 2x2 + ... (Consider up to term x2 only)

= 1 + (n - 1)x + [n(n - 1)/2 - n + 2]x2 + ...

From the given:

n - 1 = 8, so n = 9

k = n(n - 1)/2 - n + 2 = 36 - 9 + 2 = 29

2009-12-17 21:42:44 補充：
因為
1 + (n - 1)x + [n(n - 1)/2 - n + 2]x^2 + ... = 1 + 8x + kx^2+ ...

comparing coeff of x, we have n - 1 = 8

參考: Myself

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