1.for a postitive integer n,when[1-(5x/2)]^n is expanded in ascending powers of x the coefficient of x is -20.
a)find as integers,
(i)the coefficient of x^2
(ii)the coefficient of x^3
2.a)use the binomial theorem to obtain the first three terms in the expansion of (1+y)^8 in ascending powers of y.
b)hence find the coefficient of x^2 in the expansopn of (1+x-x^2)^8 in ascending powers of x.
Geometric progression
2007-03-04 6:40 am
回答 (1)
2007-03-04 7:18 am
✔ 最佳答案
1)(a)For coefficient of x, we havenC1*(-5/2) = -20
n*-2.5 = -20
n = 8
(i)coeffcient of x^2 = 8C2*(-2.5^2)
= (8*7/2/1) * 6.25 = 175
(ii)coeffcient of x^3 = 8C3*(-2.5^3)
= (8*7*6/3/2/1) * -15.625 = -875
2)(a)(1+y)^8 = 1+ 8y + 28y^2 + ...
2)(b)Let y = x-x^2
By (a), we have (1+y)^8 = 1+ 8y + 28y^2 + ...
Thus, (1+(x-x^2))^8
= 1+ 8(x-x^2) + 28(x-x^2)^2 + ...
= 1 + 8x - 8x^2 + 28x^2 + ...
Thus, coefficient of x^2 = -8 + 28 = 20
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