# The 5th term of a geometric sequence is 45 and the 8th term is 360. How do you determine the 20th term?

Feb 17, 2016

$1474560$

#### Explanation:

First we need to find the ratio, $r$ and the first term:$a$.
We know that $a \cdot {r}^{5} = 45$ and $a \cdot {r}^{8} = 360$.

To determine $r$ can divide the 8th term by the 5th term like so:

$\frac{a {r}^{8}}{a {r}^{5}} = \frac{360}{45}$

The $a$s will cancel and the rest will simplify to:

${r}^{3} = 8$
$\to r = 2$

To find $a$ simply substitute $r$ into either of the known terms, i.e:

$a \cdot {2}^{5} = 45 \to a = \frac{45}{2} ^ 5 = \frac{45}{32}$

Now we can find the twentieth term:

$a {r}^{20} = \frac{45}{32} \cdot {2}^{20} = 1474560$