The first term of a geometric sequence is 5, and the third term is 16/5. How do you find the fifth term?

Dec 9, 2015

fifth term is $\frac{256}{125}$

Explanation:

For a geometric sequence
$\textcolor{w h i t e}{\text{XXX}} {a}_{i + 1} = {a}_{i} \cdot r$ for $i > 1$ and some constant $r$
$\textcolor{w h i t e}{\text{XXX}} {a}_{i + 2} = {a}_{i} \cdot {r}^{2}$

Given
$\textcolor{w h i t e}{\text{XXX}} {a}_{1} = 5$
and
$\textcolor{w h i t e}{\text{XXX}} {a}_{3} = \frac{16}{5}$

$\textcolor{w h i t e}{\text{XXX}} = 5 \cdot {r}^{2}$

$\textcolor{w h i t e}{\text{XXX}} {r}^{2} = \frac{16}{25}$

$\textcolor{w h i t e}{\text{XXX}} {r}^{4} = \frac{{16}^{2}}{{25}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} {a}_{5} = {a}_{1} \cdot {r}^{4} = \frac{\cancel{5} \cdot {16}^{2}}{{\cancel{25}}_{5} \cdot 25} = \frac{256}{125}$