If the sum of the coefficients of all even powers of x in the product
(1 + x + x² + ....+ x²ⁿ)(1 - x + x² - x³ + ...... + x²ⁿ) is 61, then n is equal to _______.

Solution

Maths sir · 4 steps

Do not expand the product. Put x = 1 and x = -1, and average the two values.

1. Even-power trick. Call the product f(x). At x = 1 every term counts +1, and at x = -1 the even powers stay positive while the odd powers turn negative. So adding f(1) and f(-1) cancels the odd powers and doubles the even ones.

   Sum of even-power coefficients = (f(1) + f(-1))/2

2. Value at one. The first factor has 2n + 1 terms, all equal to 1, so it gives 2n + 1. The second factor alternates 1 - 1 + 1 - … + 1 over an odd number of terms, so it gives 1. So f(1) = 2n + 1.
3. Value at minus one. Now the roles swap. The first factor becomes 1 - 1 + 1 - … + 1 = 1, and every term of the second factor becomes +1, giving 2n + 1. So f(-1) = 2n + 1.
4. So. The sum is ((2n + 1) + (2n + 1))/2 = 2n + 1 = 61, so 2n = 60 and n = 30.

The idea to keep. For the sum of even-power coefficients use (f(1)+f(-1))/2. For odd powers use (f(1)-f(-1))/2. Substituting ± 1 beats expanding every time.

Answer from the NTA final answer key. Solution by PrepWiser.