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 + &#8230; + x 2 n 1 - x + x 2 - x 3 + &#8230; + x 2 n is 61 , then n is equal to

Aicharya · Accepted Answer

Let 1 - x + x 2 . . 1 + x + x 2 &#8230; = a 0 + a 1 &#215; a 2 x 2 + &#8230; Put x = 1 , 1 2 n + 1 = a 0 + a 1 + a 2 + . . a 2 n &#160; &#160; &#8230; 2 Put x = - 1 2 n + 1 &#215; 1 = a 0 - a 1 + a 2 + . . a 2 n &#160; &#160; &#8230; 2 From 1 + 2 4 n + 2 = 2 a 0 + a 2 + &#8230; = 2 &#215; 61 &#8658; 2 n + 1 = 61 &#8658; n = 30

If the sum of the coefficients of all even powers of x in the product 1 + x + x 2 + … + x 2 n 1 - x +…

Solution

If the sum of the coefficients of all even powers of x in the product 1 + x + x 2 + &#8230; + x 2 n 1 - x +…

Solution

If the sum of the coefficients of all even powers of x in the product 1 + x + x 2 + … + x 2 n 1 - x +…