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?

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Answer 1 · 2015-11-29T20:26:05

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

$r=-sqrt(7/5)$

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

hope that helped

Answer 2 · 2015-11-29T20:31:26

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

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

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.