Question

Use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 8 and the 8th term is 18?

12
15
7
5

Answer 1

If the common ratio is positive, then the `7`th term is `12`.

If not, then it would be `-12`.

Explanation:

In a geometric sequence of positive terms, the middle term of three consecutive terms is the geometric mean of the first and third.

Given two positive numbers `a` and `b`, their geometric mean is:

`sqrt(ab)`

In our example we find that the geometric mean of `8` and `18` is:

`sqrt(8*18) = sqrt(144) = 12`

So, assuming the common ratio of the geometric sequence is positive, the `7`th term is `12`.

Note that the `7`th term could also be `-12`, if the common ratio was `-3/2` instead of `3/2`.