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1) (1 + 4x)ⁿ
= 1 + n(4x) + (nC2)(4x)² + ...
= 1 + 4nx + (n(n-1)/2)16x² + ...依題意,
10(4n) = (n(n-1)/2)16
5n = n(n-1)
n(n - 6) = 0
n = 6 或 n = 0 (捨)
2a) (1 + 8x)ⁿ
= 1 + n(8x) + (nC2)(8x)² + ...
= 1 + 8nx + (n(n-1)/2)64x² + ...
= 1 + 8nx + 32n(n-1)x² + ...
∴
a = 8n
b = 32n(n-1)
b)b = 2880 = 32n(n-1)
90 = n(n-1)
n² - n - 90 = 0
(n - 10)(n + 9) = 0
n = 10 或 n = - 9 (捨)a = 8n = 80
3a) (1 - ax)⁶ + (1 - 3x)ⁿ
= (1 + 6(- ax) + (6C2)(- ax)² + ...) + (1 + n(- 3x) + (nC2)(- 3x)² + ...)
= (1 - 6ax + 15a²x² + ...) + (1 - 3nx + (n(n-1)/2)9x² + ...)
= 2 - (6a+3n)x + (15a² + (9/2)n(n-1))x² + ...
b)- (6a+3n) = - 21 ...(1)
{
15a² + (9/2)n(n-1) = 105 ...(2)由 (1) ,
n = 7 - 2a , 代入(2) :15a² + (9/2)(7-2a)(7-2a-1) = 105
15a² + (9/2)(7-2a)(6-2a) = 105
15a² + (9/2)(42 - 12a - 14a + 4a²) = 105
5a² + 3(21 - 13a + 2a²) = 35
11a² - 39a + 28 = 0
(11a - 28)(a - 1) = 0
a = 28/11 (捨) 或 a = 1n = 7 - 2a = 5