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What could  be the closed form of the generating function for the sequence {an} where

an=2n+3 for n=0,1,2,3.... ?
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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 + $\frac{2}{1-x}$

$\therefore$ $g(x)$ = $\frac{1}{1-x}$ + $\frac{2}{(1-x)^{2}}$

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

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

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