Geometric Sequences as Linear Recurrences

$\textbf{Geometric Progressions G.P.}$

A geometric progression starts with some number, called initial term and generally taken as $a$, and further terms are obtained by multiplying the previous term by some constant $r$.

Thus a geometric progression is a sequence of the form $a,ar,ar^2,ar^3,....$.

$\bullet \quad n^{th}$ term of a geometric progression is given by $t_n=ar^{n-1}$.

$\bullet \quad$Sum of $n$ terms of a geometric progression is given by $$S_n=\dfrac{a(r^n-1)}{r-1}$$

See that here, if $| r | <1$, then the G.P. will be a decreasing G.P., as each term will be smaller than previous term.

For decreasing G.P., the sum of infinite terms is given by $s_\infty = \dfrac{a}{1-r}$,

Because in the above obtained formula, if $|r|<1$, as $n\rightarrow \infty , \dfrac{r^n-1}{r-1} \rightarrow \dfrac{1}{1-r}$

This gives for $|r|<1$, $\boxed{S_\infty =\dfrac{a}{1-r}}$ for G.P.