Question

How do you find the fifth term in a geometric sequence in which the fourth term is 5, the sixth term is 7, and the common ratio is negative?

Answer 1

`r= -sqrt("7th term"/"5th term")`

Explanation:

`r=-sqrt(7/5)`

5th term `=5xx-sqrt(7/5)~~-5.916`

hope that helped

Answer 2

The fifth term will be a geometric mean of the fourth and sixth term.

Since the common ratio is negative it will be `-sqrt(35)`

Explanation:

The general form of a term of a geometric sequence is:

`a^n = a r^(n-1)`

Where `a` is the initial term and `r` the common ratio.

We are given `a_4 = a r^3 = 5` and `a_6 = a r^5 = 7`

So `r^2 = (a r^5) / (a r^3) = a_6 / a_4 = 7 / 5`

So `r = -sqrt(7/5)`

Then `a_5 = r a_4 = 5 (-sqrt(7/5)) = -sqrt(7)sqrt(5) = -sqrt(35)`

Or just taking the geometric mean of `a_4` and `a_6`...

`a_5 = +-sqrt(a_4 * a_6) = +-sqrt(35)`

and we need to use the negative square root to get a negative common ratio.