Let the generating function G(x) for ${a_{k}}$ be $\sum_{k=0}^{\infty }a_{k}.x^k$
$G(x)=\sum_{k=0}^{\infty }a_{k}.x^k$ $........(1)$
multiply both side $x$, we get
$x.G(x)=\sum_{k=0}^{\infty }a_{k}.x^{k+1}$ $............(2)$
$1−2$
($1−x).G(x)=\sum_{k=0}^{\infty }a_{k}.x^k - \sum_{k=0}^{\infty }a_{k}.x^{k+1}$
$=a_{0}+\sum_{k=1}^{\infty }a_{k}.x^k - \sum_{k=1}^{\infty }a_{k-1}.x^{k}$
$=a_{0}+ \sum_{k=1}^{\infty }(a_{k}-a_{k-1}).x^{k}$
$=a_{0}+\sum_{k=1}^{\infty }2.x^k $
$=a_{0}+2.\sum_{k=1}^{\infty }x^k $
$=a_{0}+2x/1-x$
$=3+2x/1-x$
$= (3-x)/1-x$
$(1-x).G(x)=(3−x)/(1−x)$
$=(3−x)/(1−x)(1-x)$
$=(3-x)/(1-x)^2$
$N.B$
1. $a_{k}=2k+3$
$a_{k−1}=2(k−1)+3=2k−2+3$
$a_{k}−a_{k−1}=2$
2. sum of infinite $GP$ is $a/1−r $, where $r$ is common ratio and $a$ is the first term
3. $a_{0}=3$
