# 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?

Nov 29, 2015

$r = - \sqrt{\text{7th term"/"5th term}}$

#### Explanation:

$r = - \sqrt{\frac{7}{5}}$

5th term $= 5 \times - \sqrt{\frac{7}{5}} \approx - 5.916$

hope that helped

Nov 29, 2015

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 ${r}^{2} = \frac{a {r}^{5}}{a {r}^{3}} = {a}_{6} / {a}_{4} = \frac{7}{5}$

So $r = - \sqrt{\frac{7}{5}}$

Then ${a}_{5} = r {a}_{4} = 5 \left(- \sqrt{\frac{7}{5}}\right) = - \sqrt{7} \sqrt{5} = - \sqrt{35}$

Or just taking the geometric mean of ${a}_{4}$ and ${a}_{6}$...

${a}_{5} = \pm \sqrt{{a}_{4} \cdot {a}_{6}} = \pm \sqrt{35}$

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