Which one of the following is the closed-form generating function for the sequence {a_n}_{n>=0} defined by a_n = 2n + 3, for n = 0, 1, 2, ...? A. (3 - x) / (1 - x)^2 B. (3 + x) / (1 - x)^2 C. (2 + 3x) / (1 - x)^2 D. (3 - x) / (1 + x)^2

GATE 2018 · Discrete Mathematics · Generating Functions · medium

Answer: (3 - x) / (1 - x)^2

1. Decompose a_n and write G(x) as a sum: G(x) = sum_{n>=0}(2n+3)x^n = 2 sum_{n>=0} n x^n + 3 sum_{n>=0} x^n.
2. Substitute known closed forms: sum_{n>=0} n x^n = x/(1-x)^2 and sum_{n>=0} x^n = 1/(1-x). So G(x) = 2x/(1-x)^2 + 3/(1-x).
3. Combine over a common denominator: 3/(1-x) = 3(1-x)/(1-x)^2. So G(x) = [2x + 3(1-x)] / (1-x)^2 = (2x + 3 - 3x) / (1-x)^2 = (3 - x) / (1-x)^2.