If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^{2}+\ldots \ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots . .+x^{2 n}\right)$ is 61 , then $n$ is equal to

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Binomial Theorem

If the sum of the coefficients of all even powers of

x

in the product

(1 + x + x^{2} + \dots\dots + x^{2 n}) (1 - x + x^{2} - x^{3} + \dots .. + x^{2 n})

is 61 , then

n

is equal to

Answer: 30

Let

P (x) = (1 + x + \dots + x^{2 n}) (1 - x + x^{2} - \dots + x^{2 n})

. Evaluating at

x = 1

and

x = - 1

both give

2 n + 1

, and adding the two picks out twice the even-power coefficients:

a_{0} + a_{2} + a_{4} + \dots = \frac{P ( 1 ) + P ( - 1 )}{2} = 2 n + 1

. Setting this equal to

61

gives

2 n + 1 = 61

, so

n = 30

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Integer answer