Write the equation for the $\mathit{n}\mathit{t}\mathit{h}$term of each geometric sequence.

$\mathbf{256}\mathbf{,}\mathbf{128}\mathbf{,}\mathbf{64}\mathbf{,}\mathbf{.}\mathbf{....}$

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

The sequence:$256,128,64,\mathrm{.....}$

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 $a_1=256$and the common ratio is:

$\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{128}{256}\\ =\frac{1}{2}\end{array}$.

In order to calculate the nth term substitute 256 for into the formula .

$\begin{array}{c}{a}_n=256\times {\left(\frac{1}{2}\right)}^{n-1}\\ =2^8\times 2^{-(n-1)}\\ =2^8\times 2^{-n+1}\\ =2^{8-n+1}\\ =2^{9-n}\end{array}$

The nth term of the series $256,128,64,\mathrm{.....}$is $a_n=2^{9-n}$.