# What is the 7th term of the geometric sequence where a1 = –4 and a5 = –1,024?

Nov 25, 2015

$- 16384$ or $16384$

#### Explanation:

The common ratio $r$ of this geometric sequence is a $4$th root of $\frac{- 1024}{- 4} = 256 = {4}^{4}$, since ${a}_{5} = {r}^{4} {a}_{1}$

The possible Real $4$th roots are $\pm 4$

In either case:

${a}_{7} = {r}^{2} {a}_{5} = 16 \cdot \left(- 1024\right) = - 16384$

So why do I say that ${a}_{7}$ may be $16384$?

The Complex numbers $\pm 4 i$ are also $4$th roots of $256$.

If $r = \pm 4 i$ then ${r}^{2} = - 16$ and we find:

${a}_{7} = {r}^{2} {a}_{5} = \left(- 16\right) \cdot \left(- 1024\right) = 16384$