# How do you write an equation for the nth term of the geometric sequence 4, 8, 16, ...?

Jan 13, 2017

${a}_{n} = 4 \cdot {2}^{n - 1}$
...but see below

#### Explanation:

It seems that currently the most common usage is that the initial value of a sequence is considered the "first" value.
That is for the given geometric sequence: 4, 8, 16, ...
$\textcolor{w h i t e}{\text{XXX}} {a}_{1} = 4$
Each term after the first increases the immediately prior term by a factor of $2$ and since there are $\left(n - 1\right)$ terms after the first for the ${n}^{t h}$ term, we have
$\textcolor{w h i t e}{\text{XXX")a_n = a_1 xx underbrace(2 xx 2 xx ... xx2)_(n" times}} = {a}_{1} \cdot {2}^{n - 1} = 4 \cdot {2}^{n - 1}$

Less common these days (but the version I learned long ago) was to call the initial value of the sequence, the ${\text{zero}}^{t h}$ value.
That is for the given sequence: 4, 8, 16, ...
$\textcolor{w h i t e}{\text{XXX}} {a}_{0} = 4$
and the equation for the ${n}^{t h}$ value would be
$\textcolor{w h i t e}{\text{XXX}} {a}_{n} = {a}_{0} \cdot {2}^{n}$ or $4 \cdot {2}^{n}$