GATE CS 2018 Q.11 (CS Q.1): Combinatorics, Counting, Recurrence Relations and Generating Functions

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 $n = 0, 1, 2,\dots$ ?

$\frac{3}{(1-x)^2}$

$\frac{3x}{(1-x)^2}$

$\frac{2-x}{(1-x)^2}$

$\frac{3-x}{(1-x)^2}$

Official answer and solution

Official answer: D
From: Official GATE 2018 answer key

Solution

Use the standard generating function series expansions for constant and linear terms.

The generating function $G(x)$ for a sequence $\{a_n\}$ is defined as:

G(x) = \sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} (2n + 3) x^n = 2 \sum_{n=0}^{\infty} n x^n + 3 \sum_{n=0}^{\infty} x^n

Recall the standard closed-form expressions:

\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}

Taking the derivative with respect to $x$ and multiplying by $x$ :

\sum_{n=0}^{\infty} n x^n = x \frac{d}{dx}\left(\frac{1}{1-x}\right) = \frac{x}{(1-x)^2}

Substitute these into $G(x)$ :

G(x) = 2 \left(\frac{x}{(1-x)^2}\right) + 3 \left(\frac{1}{1-x}\right) = \frac{2x + 3(1-x)}{(1-x)^2} = \frac{2x + 3 - 3x}{(1-x)^2} = \frac{3 - x}{(1-x)^2}

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2018 Computer Science and Information Technology (CS)
Question: 11 of 65, printed as CS Q.1
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source