# How do you find the 8th term in this geometric sequence 8, 4, 2, 1, ...?

Dec 18, 2015

Find the common ratio and use it to find that the eighth term is
$\frac{1}{16}$

#### Explanation:

Given a geometric series $\left(a {r}^{n}\right)$ with initial term $a$ and common ratio $r$, we may find $r$ by dividing any term after the first by the prior term, as
$\frac{a {r}^{k}}{a {r}^{k - 1}} = r$

Thus in the given sequence, dividing the second term by the first gives
$r = \frac{4}{8} = \frac{1}{2}$

The ${n}^{t h}$ term in the sequence is $a {r}^{n - 1}$. Thus, as the given sequence has an initial term $a = 8$, the eighth term is

$a {r}^{7} = 8 {\left(\frac{1}{2}\right)}^{7} = \frac{1}{16}$