關於數學M1 (2x^2+1)^
2010-10-30 11:01 pm
In the expansion of (2x^2+1)^n in ascending powers of x, where n is a positive integer, the coefficient of the third term is 60.Find the value of n and the coefficient of x^8
回答 (1)
2010-10-30 11:51 pm
✔ 最佳答案
(2x^2 + 1)^n= (1 + 2x^2)^n
= 1 + 2nx^2 + n(n-1)/2 * (2x^2)^2 + n(n-1)(n-2)/6 * (2x^2)^3 + n(n-1)(n-2)(n-3)/24 * (2x^2)^4 + ...
= 1 + 2nx^2 + 2n(n-1)x^4 + 4n(n-1)(n-2)/3 * x^6 + 2n(n-1)(n-2)(n-3)/3 * x^8 + ...
Coef. of the third term = 60
2n(n-1) = 60
n^2 - n - 30 = 0
(n-6)(n+5) = 0
n = 6 or n = -5(rejected)
The coef. of x^8
= 2(6)(6-1)(6-2)(6-3)/3
= 240
參考: Knowledge is power.
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