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

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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

$- 16384$ $-16384$ or $16384$ $16384$

Explanation:

The common ratio $r$ $r$ of this geometric sequence is a $4$ $4$ th root of $\frac{- 1024}{- 4} = 256 = 4^{4}$ $(-1024)/(-4) = 256 = 4^4$ , since $a_{5} = r^{4} a_{1}$ $a_5 = r^4 a_1$

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

In either case:

$a_{7} = r^{2} a_{5} = 16 \cdot (- 1024) = - 16384$ $a_7 = r^2 a_5 = 16*(-1024) = -16384$

So why do I say that $a_{7}$ $a_7$ may be $16384$ $16384$ ?

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

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

$a_{7} = r^{2} a_{5} = (- 16) \cdot (- 1024) = 16384$ $a_7 = r^2 a_5 = (-16)*(-1024) = 16384$

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

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Explanation:

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