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

The sequence:$−1,1,−1,1,\mathrm{.....}$

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$a_1=−1$and the common ratio is:

$\begin{array}{c}r=\text{ â¶Ä‰}\frac{a_2}{a_1}\\ =\frac{1}{(−1)}\\ =−1\end{array}$

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

$\begin{array}{c}{a}_n=−1×{(−1)}^{n−1}\\ ={(−1)}^1×{(−1)}^{n−1}\\ ={(−1)}^{1+n−1}\\ ={(−1)}^n\end{array}$

The $nth$term of the series $−1,1,−1,1,\mathrm{.....}$is$a_n={(−1)}^n$.