Question

Find the generating function, the general term, and the sum to n terms of the recurring series $1-7x-x^2-43x^3-....$.

Answer 1

Let the scale of relation be $1-px-q{x}^2;$ then $-1+7p-q=0,-43+p+7q=0;$
whence $p=1,q=6;$ and the scale of relation is $1-x-6x^2$.
Let S denote the sum of the series; then
$S=1-7x-x^2-43x^3-....$
$-xS=-x+7x^2+x^3+....$
$-6x^{2}S=-6x^2+42x^3+....$
$\therefore (1-x-6x^2)S=1-8x,$
$S=\frac{1-8x}{1-x-6x^2};$
Which is the generating function.