# What is the 5th term in the following geometric sequence 2, -14, 98, -686?

Dec 11, 2016

${a}_{5} = 4802$.

#### Explanation:

In a geometric sequence, each new term is the product of the previous term and a fixed multiplier $r$. (This is in contrast to an arithmetic sequence, where each new term is the sum of the previous term and a fixed additive value $a$.)

To find the multiplier $r$ of a geometric sequence, simply take the ratio of any two consecutive terms already in the sequence.

$r = \frac{{a}_{n}}{{a}_{n - 1}}$

For example, if we take the ratio of the 2nd-to-1st terms, we get:

$r = \frac{{a}_{2}}{{a}_{1}} = \text{-14"/2="-7}$

Then, since we know the 4th term is -686, all we need to do is multiply this by our ratio $r$ to get the 5th term:

${a}_{5} = r {a}_{4} = \left(\text{-7")("-686}\right) = 4802$.

## Bonus:

In general, a geometric sequence is written as

$a , a r , a {r}^{2} , a {r}^{3} , \ldots$

where $a$ is the starting value of the sequence and $r$ is the common ratio between successive terms. If you know both the first term and the common ratio, you can find the ${n}^{\text{th}}$ term (${a}_{n}$) by solving ${a}_{n} = a {r}^{n - 1} .$ (In this case, we would get

${a}_{5} = a {r}^{5 - 1} = 2 \cdot {\left(\text{-7}\right)}^{4} = 2 \cdot 2401 = 4802 ,$

same as before.)