Question

Find the generating function and the general term of the following series: $2-x+5x^2-7x^3+.....$

Answer 1

To find the generating function and the general term of the following series

$2-x+5x^2-7x^3+......$

Let , $1-px-q{x}^2$ be the scale of relation ;

Whence $p=-1,,q=-2$

$\therefore (1+x-2x^2)=2+x$ ;

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

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

$\therefore$ the general term

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