an = 2n+3 for n = 0,1,2,3...
We can see that
a0 = 3
a1 = 5
a2 = 7
a3 = 9 and so on
And we know,
$g(x)$ = $\sum_{r = 0}^{\infty }a_{r}x^{r}$
$\therefore g(x)$ = $3 + 5x + 7x^{2} + 9x^{3} + .....$ ...(1)
Now, $x$.g($x$) = $3x + 5x^{2} + 7x^{3} + 9x^{4} + ....$ ...(2)
(1) $-$ (2) ,
$g(x).(1-x)$ = $3 + 2x + 2x^{2} + 2x^{3} + 2x^{4} + ....$
= $1 + 2 + 2x + 2x^{2} + 2x^{3} + 2x^{4} + ....$
= 1 + $\frac{2}{1-x}$
$\therefore$ $g(x)$ = $\frac{1}{1-x}$ + $\frac{2}{(1-x)^{2}}$
= $\frac{1-x}{(1-x)^{2}}$ + $\frac{2}{(1-x)^{2}}$
= $\frac{3-x}{(1-x)^{2}}$