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

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Answer 1 · 2015-11-25T00:25:16

$-16384$ or $16384$

The common ratio $r$ of this geometric sequence is a $4$ th root of $(-1024)/(-4) = 256 = 4^4$ , since $a_5 = r^4 a_1$

The possible Real $4$ th roots are $+-4$

In either case:

$a_7 = r^2 a_5 = 16*(-1024) = -16384$

So why do I say that $a_7$ may be $16384$ ?

The Complex numbers $+-4i$ are also $4$ th roots of $256$ .

If $r = +-4i$ then $r^2 = -16$ and we find:

$a_7 = r^2 a_5 = (-16)*(-1024) = 16384$