# How do you use the geometric sequence formula to find the nth term?

Jan 13, 2015

A geometric sequence is always of the form
${t}_{n} = {t}_{\text{n-1}} \cdot r$

Every next term is $r$ times as large as the one before.

So starting with ${t}_{0}$ (the "start term") we get:
${t}_{1} = r \cdot {t}_{0}$
${t}_{2} = r \cdot {t}_{1} = r \cdot r \cdot {t}_{0} = {r}^{2} \cdot {t}_{0}$
......
${t}_{n} = {r}^{n} \cdot {t}_{0}$

${t}_{n} = {r}^{n} \cdot {t}_{0}$
${t}_{0}$ being the start term, $r$ being the ratio

Extra:
If $r > 1$ then the sequence is said to be increasing
if $r = 1$ then all numbers in the sequence are the same
If $r < 1$ then the sequence is said to be decreasing ,
and a total sum may be calculated for an infinite sequence:
sum $\sum = {t}_{0} / \left(1 - r\right)$

Example :
The sequence $1 , \frac{1}{2} , \frac{1}{4} , \frac{1}{8.} . .$
Here the ${t}_{0} = 1$ and the ratio $r = \frac{1}{2}$
Total sum of this infinite sequence:
$\sum = {t}_{0} / \left(1 - r\right) = \frac{1}{1 - \frac{1}{2}} = 2$