# The fifth term of a geometric sequence is 48 and the ninth term is 768. What is the first term?

Nov 6, 2015

The first term is 3.

#### Explanation:

The general form of a geometric sequence with the first term $a$ is $a , a r , a {r}^{2} , a {r}^{3} , \ldots$ where $r$ is the common ratio between terms.

Note that the general form for the $n$th term is $a {r}^{n - 1}$.
Using that with our knowledge of the fifth and ninth terms, we have $a {r}^{4} = 48$ and $a {r}^{8} = 768$

We can now eliminate $a$ by dividing to obtain
$\frac{a {r}^{8}}{a {r}^{4}} = \frac{768}{48}$
$\implies {r}^{4} = 16$

We could solve for $r$ by taking the fourth root of 16, but that is not necessary for our goal of finding a.

Now that we have ${r}^{4}$ we can divide once again to find $a$.
$\frac{a {r}^{4}}{r} ^ 4 = \frac{48}{16}$

$\implies a = 3$