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# First three terms in expansion of (1 − 2x)2(1 + ax)6, in ascending powers of x, are 1 − x + bx2. Find a and b?

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• First three terms in expansion of (1 − 2x)2(1 + ax)6, in ascending powers of x, are 1 − x + bx2. Find a and b?

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Answer #1 | 07/02 2014 06:13
a = 1/2; b = - 17/4 f(x) = (1 − 2x)^2(1 + ax)^6 f '(x) = 2( -2) (1 - 2x) (1 + ax)^6 + (1 − 2x)^2 6a(1 + ax)^5 = = 2(1 - 2x)(1 + ax)^5 ( -2(1 + ax) + 3a(1 - 2x) ) = f '(x) = 2 (2 x - 1) (8 a x - 3 a + 2) (a x + 1)^5 f ''(x) = 2 (112 a^2 x^2 + 28 a x (2 - 3 a) + 15 a^2 - 24 a + 4) (a x + 1)^4 f(0) = 1 f '(0) = 2 (3 a - 2) f ''(0) = 2 (15 a^2 - 24 a + 4) Taylor expansion is 1 + 2x(3a - 2) + x^2 (15a^2 - 24a + 4) coefficient of x is 2(3a - 2) = - 1 a = 1/2 plug in the coefficient of x^2 15(1/2)^2 - 24(1/2) + 4 = - 17/4
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Answer #2 | 07/02 2014 05:55
First three terms in expansion of (1 - 2x)^2(1 + ax)^6, in ascending powers of x, are 1 - x + bx2. 1 - 4x + 4x^2 b = 4 Find a and b
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