If A = Σₙ₌₁^∞ 1/( 3 + ( - 1)ⁿ )ⁿ and B = Σₙ₌₁^∞ ( - 1)ⁿ/( 3 + ( - 1)ⁿ )ⁿ , then A/B is equal to :

Solution

Maths sir · 4 steps

Split the sum into odd n and even n; then each part is a plain GP with its own fixed base.

1. Fix the base. For odd n, 3 + (-1)ⁿ = 2, so the term is 1/2ⁿ. For even n, 3 + (-1)ⁿ = 4, so the term is 1/4ⁿ.
2. Sum each part. Odd terms: 1/2 + 1/8 + 1/32 + … = (1/2)/(1 - 1/4) = 2/3. Even terms: 1/16 + 1/256 + … = (1/16)/(1 - 1/16) = 1/15.
3. Attach the signs. In A every term is positive. In B the factor (-1)ⁿ makes the odd terms negative and leaves the even terms positive.

   A = 2/3 + 1/15 = 11/15, B = -2/3 + 1/15 = -9/15

4. So. A/B = (11/15)/(-9/15) = -11/9, which is option (c).

The idea to keep. When (-1)ⁿ sits inside a series, split it into odd and even n. Each half becomes an ordinary infinite GP that you can sum with a/(1-r).

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