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標題:

關於數學M1 (2x^2+1)^

發問:

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

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• 數學 10點@1@

• NDS問題(岩岩買NDS)
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• 數學 10點@1@

 

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(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

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