Question

How do you use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 12 and the 8th term?

Answer 1

If the common ratio is positive, then:

`a_7 = sqrt(a_6 * a_8)`

Explanation:

If you are given the `6th` and `8th` terms of a geometric series then there are two possibilities for the `7th` term, namely `+-sqrt(a_6 * a_8)`

If you are told that the common ratio is positive or that all of the terms of the sequence are positive then `a_7 = sqrt(a_6 * a_8)` is the geometric mean of `a_6` and `a_8`. Otherwise it could be `-sqrt(a_6 * a_8)`.

Why does this work?

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.

If `a, r > 0` then:

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

In fact, in general:

`sqrt(a_n * a_(n+2)) = a_(n+1)`

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In the given example `a_6 = 12` and let us suppose `a_8 = 48`

Then `a_7 = sqrt(a_6 * a_8) = sqrt(12 * 48) = sqrt(24*24) = 24`