Series (basic math)

In series with a hanging 1, times through by r

$x_n = r^{n-1} + r^{n-2} + ... + r + 1$

$rx_n = r^{n} + r^{n-1} + r$

and then subject the original series to simplify

$x_1 = a$,   $x_{n+1} = rx_n + d$,    $(n=1,2,3,...)$

$x_1 = a$

$x_2 = ra + d$

behaviour of $r^n$ in the limit

geometric series
sum; $a = ar + ar^2 + ... + ar^n$

$a + ar + ar^2 + ... + ar^n = \frac{1-r^{n+1}}{1-r}$, $r \ne 1$