If the sum of the coefficients of all even powers of x in the product (1 + x + x^(2) + ....+ x^(2n))(1 - x + x^(2) - x^(3) + ...... + x^(2n)) is 61, then n is equal to _______.

(1 + x + x^(2) + ....+ x^(2n))(1 - x + x^(2) - x^(3) + ...... + x^(2n)) = a_(0) + a_(1)x + a_(2)x^(2) + ….. put x = 1 (2n + 1) x 1 = a_(0) + a_(1) + a_(2) + …… (1) put x = –1 1 x (2n + 1) = a_(0) – a_(1) + a_(2)+ …….. (2) Adding (1) and (2) 4n + 2 = 2(a_(0) + a_(2) + ….. ) => 4n + 2 = 2 x 61 => n = 30

If the sum of the coefficients of all even powers of x ... | null (Mathematics)

(1 + x + x² + ....+ x²ⁿ)(1 - x + x² - x³ + ...... + x²ⁿ)

= a₀ + a₁x + a₂x² + …..

put x = 1

(2n + 1)

\times

1 = a₀ + a₁ + a₂ + …… (1)

put x = –1

1

\times

(2n + 1) = a₀ – a₁ + a₂+ …….. (2)

Adding (1) and (2)

4n + 2 = 2(a₀ + a₂ + ….. )

\Rightarrow

4n + 2 = 2

\times

\Rightarrow

n = 30

Explanation

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