Write the equation for the nth term of each geometric sequence.

$\mathbf{3}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{27}\mathbf{,}\mathbf{.}\mathbf{....}$

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For example, the sequence $2,6,18,54,...$is a geometric progression with common ratio 3.

The ratio of $2^{nd}$term and first term is called the common ratio.

The sequence: $3,9,27,\mathrm{.....}$

Each term in a geometric sequence can be expressed in terms of the first term $a_1$and the common ratio $r$. Since each succeeding term is formulated from one or more previous terms, this is a recursive formula.

First-term$a_1=3$and the common ratio is:

$\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{9}{3}\\ =3\end{array}$

In order to calculate the $nth$term substitute 3 for $a_1$into the formula $a_n=a_1{r}^{n-1}$.

$\begin{array}{c}{a}_n=3\times {\left(3\right)}^{n-1}\\ ={\left(3\right)}^{1+n-1}\\ ={\left(3\right)}^n\end{array}$

The nth term of the series $3,9,27,\mathrm{.....}$is $a_n={\left(3\right)}^n$.