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GATE 2018 CS – Question 11

Engineering Mathematics · Discrete Mathematics: Combinatorics (Counting, Recurrence Relations, Generating Functions) · 1 mark · Multiple choice

Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n=2n+3$ for all $n=0,1,2,\dots$?

  1. $\dfrac{3}{(1-x)^2}$
  2. $\dfrac{3x}{(1-x)^2}$
  3. $\dfrac{2-x}{(1-x)^2}$
  4. $\dfrac{3-x}{(1-x)^2}$

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Correct answer: (D) $\dfrac{3-x}{(1-x)^2}$

Explanation

$\sum2nx^n=\frac{2x}{(1-x)^2}$ and $\sum3x^n=\frac{3}{1-x}=\frac{3(1-x)}{(1-x)^2}$. The sum is $\frac{2x+3-3x}{(1-x)^2}=\frac{3-x}{(1-x)^2}$.