What is the 8th term of the geometric sequence –1, 4, –16, …?

Nov 9, 2015

The 8th term is $- 16384$

Explanation:

A geometric sequence is a sequence of the form
$a , a r , a {r}^{2} , a {r}^{3} , \ldots$
where $a$ is the initial value and $r$ is a common ratio between terms.

Because of this, given any two successive terms of the sequence, you can find $r$ by dividing the later term by the previous one:
$\frac{a {r}^{n}}{a {r}^{n - 1}} = r$

In the given sequence, then, we can find the ratio by dividing the second term by the first:
$\frac{4}{-} 1 = - 4$

Finally, to obtain the 8th term in the sequence, we can either calculate the term by multiplying successive terms by the ratio repeatedly, or directly as $\left(- 1\right) {\left(- 4\right)}^{7}$

In either case, we arrive at the 8th term being $- 16384$