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?

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Answer 1 · 2015-11-19T07:21:05

Find that $a_7 = sqrt(a_6a_8)$ , hence $a_7 = 36$

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:

$a_7 = a r^6 = sqrt(a r^6 * a r^6) = sqrt(a r^5 * a r^7) = sqrt(a_6 a_8)$

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:

$a_7 = sqrt(a_6 a_8) = sqrt(6*6^3) = sqrt(6^2*6^2) = 6^2 = 36$