If the sum of the coefficients of all even powers of x in the product 1 + x + x 2 + … + x 2 n 1 - x + x 2 - x 3 + … + x 2 n is 61 , then n is equal to

Question

If the sum of the coefficients of all even powers of x in the product 1 + x + x 2 + … + x 2 n 1 - x + x 2 - x 3 + … + x 2 n is 61 , then n is equal to,

MARKS App · Accepted Answer

Let 1 - x + x 2 . . 1 + x + x 2 … = a 0 + a 1 × a 2 x 2 + … Put x = 1 , 1 2 n + 1 = a 0 + a 1 + a 2 + . . a 2 n … 2 Put x = - 1 2 n + 1 × 1 = a 0 - a 1 + a 2 + . . a 2 n … 2 From 1 + 2 4 n + 2 = 2 a 0 + a 2 + … = 2 × 61 ⇒ 2 n + 1 = 61 ⇒ n = 30

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