# How do you write a rule for the nth term of the geometric term given the two terms a_1=1, a_3=9?

May 21, 2017

There are two possible common ratios and corresponding sequences, given by the formula:

${a}_{n} = {3}^{n - 1} \text{ }$ or $\text{ } {a}_{n} = {\left(- 3\right)}^{n - 1}$

#### Explanation:

The general formula for a term of a geometric sequence is:

${a}_{n} = a {r}^{n - 1}$

where $a$ is the initial term and $r$ the common ratio.

In our example, we find:

$9 = {a}_{3} / {a}_{1} = \frac{a {r}^{2}}{a} = {r}^{2}$

Hence $r = \pm 3$, giving two possible sequences.

The rule for the n$t h$ term of this sequence can be written:

${a}_{n} = {3}^{n - 1}$

or:

${a}_{n} = {\left(- 3\right)}^{n - 1}$