Question

Find the generating function for the sequence {7, 8, 9, 10,.

Answer 1

11, 12, 13, 14,.......like this it will be go on.

Answer 2

So, The generating function for the sequence {$7,8,9,10,.........$ is $\frac{1}{(1-x)^{2} }$

Step-by-step explanation:

Given,

Find the generating function for the sequence {$7,8,9,10,.........$

The generating function for the sequence {$7,8,9,10,.........$ is $\frac{1}{(1-x)^{2} }$

Take a second derivative:  $\frac{2}{(1-x)^{3} } =2+6x+12x^{2} +20x^{3} +.......$

So,

$\frac{1}{(1-x)^{3} } =1+3x+6x^{2} +10x^{3} +.......$ is a generating function for the triangular numbers $1,3,6,10.....$ (Although here we have $a_{0}=1$ while $T_{0}=0$ usually.)

∴ The generating function for the sequence {$7,8,9,10,.........$ is $\frac{1}{(1-x)^{2} }$