# How do you use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 6 and the 8th term is 216?

Nov 19, 2015

Find that ${a}_{7} = \sqrt{{a}_{6} {a}_{8}}$, hence ${a}_{7} = 36$

#### Explanation:

The general term of a geometric sequence can be written:

${a}_{n} = a {r}^{n - 1}$

where $a$ is the initial term and $r$ the common ratio.

So ${a}_{6} = a {r}^{5}$, ${a}_{7} = a {r}^{6}$ and ${a}_{8} = a {r}^{7}$

So we find:

${a}_{7} = a {r}^{6} = \sqrt{a {r}^{6} \cdot a {r}^{6}} = \sqrt{a {r}^{5} \cdot a {r}^{7}} = \sqrt{{a}_{6} {a}_{8}}$

That is: ${a}_{7} = \sqrt{{a}_{6} {a}_{8}}$

In other words, ${a}_{7}$ is the geometric mean of ${a}_{6}$ and ${a}_{8}$

In our particular example ${a}_{6} = 6$ and ${a}_{8} = 216 = {6}^{3}$,

So:

${a}_{7} = \sqrt{{a}_{6} {a}_{8}} = \sqrt{6 \cdot {6}^{3}} = \sqrt{{6}^{2} \cdot {6}^{2}} = {6}^{2} = 36$