# How do you identify 19th term of a geometric sequence where a1 = 14 and a9 = 358.80?

Dec 29, 2015

An explanation is given below.

#### Explanation:

We are to find $19$th term of Geometric Sequence
Given ${a}_{1} = 14$ and ${a}_{9} = 358.80$

The general term of a Geometric Sequence is given by

${a}_{n} = a \cdot {r}^{n - 1}$
Where $a$ is the first term also known as ${a}_{1}$ and $r$ is the common ratio.

We have ${a}_{1}$ if we get $r$ we can easily find ${a}_{19}$ by using $19$ for $n$

Let us start by writing the given term using $r$

${a}_{9} = a \cdot {r}^{8}$

If we divide ${a}_{9}$ by ${a}_{1}$ we would get an equation in $r$

$\frac{a {r}^{8}}{a} = \frac{358.80}{14}$

${r}^{8} = 25.628571428571428571428571428571$
Taking $8$th root.

$r = \sqrt[8]{25.628571428571428571428571428571}$
r=1.4999975504465127405341330547934"

$r \approx 1.5$

${a}_{19} = 14 {\left(1.5\right)}^{19}$

a_19 = 31,035.729480743408203125"#
${a}_{19} \approx 31035.73$