Question

Find the generating function and the general term of the following series: $7-6x+9x^2+27x^4+....$

Answer 1

$A=7-6x+9x^2+27x^4+...$

$4x=7x-6x^2+9x^3+27x^5$

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$=7+x+3x^2+9x^3+27x^4+...$

$=7+x(1+3x+9x^2+27x^3+...)$

$=7+x(1-3x)-1$

$=7+\frac{x}{1-3x}$

$A(1+x)=\frac{7-20x}{1-3x}$

$A=\frac{7-20x}{(1-3x)(1+x)}$

So generating function is $\frac{7-20x}{(1-3x)(1+x)}$