GATE Overflow Test Series | Mixed Subjects | Test 1 | Question: 16

Question

The generating function corresponding to $4,5,7,10,14,19,25,\dots $ is ?

• A. $\frac{4}{1-x} + \frac{x}{(1-x)^3}$
• B. $\frac{2}{1-x} + \frac{x}{(1-x)^3}$
• C. $\frac{4}{1-x} + \frac{x}{(1-x)^2}$
• D. $\frac{x}{(1-x)^3}$

Answer 1

Let the required generating function be $A$

$A = 4 + 5x + 7x^2 +10x^3 \ldots$

$xA = 4x + 5x^2 + 7x^3 + \ldots$

$A - xA = 4 + ( 0 + x + 2x^2 + 3x^3 +4x^4 + \ldots)$

$\qquad = 4 + \dfrac{x}{(1-x)^2}$

$\implies A = \dfrac{4}{1-x} +\dfrac{x}{(1-x)^3} $

Correct Answer: Option A