Question

Find the generating function and the general term of the following series: $2+3x+5x^2+9x^3+....$

Answer 1

Let

$A=2+3x+5x^2+3x^3+...$

$Ax=2x+3x^2+5x^3+9x^4$

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$=2+x+2x^2+4x^2+3x^4$

$=2+x(1+2x+4x^2+3x^3+...)$

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

$=2+\frac{x}{1-2x}$

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

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

So generating fuction is

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

General term can be given as

$(1+2^n)x^n$ where $nϵ[0,\infty ]$