# How do you find the next three terms in the geometric sequence -16, 4, , , ... ?

Nov 6, 2015

Find the common ratio $r$ between terms, and multiply by it repeatedly to obtain $- 1 , \frac{1}{4} , - \frac{1}{16}$ as the next three terms in the sequence.

#### Explanation:

The general form for a geometric sequence with the first term $a$ is $a , a r , a {r}^{2} , a {r}^{3} , \ldots$ where $r$ is a common ratio between terms.

As the first two terms of the geometric sequence given are $- 16$ and $4$, we have $a = - 16$ and $a r = 4$.

Then, to find $r$, we simply divide the second term by the first to obtain
$\frac{a r}{a} = \frac{4}{- 16}$
$\implies r = - \frac{1}{4}$

Thus the next three terms in the sequence will be
$a {r}^{2} = 4 \cdot \left(- \frac{1}{4}\right) = - 1$
$a {r}^{3} = - 1 \cdot \left(- \frac{1}{4}\right) = \frac{1}{4}$
$a {r}^{4} = \frac{1}{4} \cdot \left(- \frac{1}{4}\right) = - \frac{1}{16}$

Nov 6, 2015

$\left({x}_{n}\right) = - 16 , 4 , - 1 , \frac{1}{4} , - \frac{1}{16} , \ldots$

#### Explanation:

Since it is a geometric sequence $\left({x}_{n}\right)$, there is a constant ratio $r = \left({x}_{n + 1} / {x}_{n}\right) = \frac{4}{-} 16 = - \frac{1}{4}$

So if $a = - 16$ is the first term ${x}_{1}$, then general term is given by ${x}_{n} = a {r}^{n - 1} = - 16 \cdot {\left(- \frac{1}{4}\right)}^{n - 1}$

Hence the 3rd term, ${x}_{3} = - 16 \cdot {\left(- \frac{1}{4}\right)}^{3 - 1} = - 1$

Similarly ${x}_{4} = \frac{1}{4} \mathmr{and} {x}_{5} = - \frac{1}{16}$