Question

Find $a_{20}$ of a geometric sequence if the first few terms of the sequence are given by $-\frac{1}{2},\frac{1}{4},-\frac{1}{8},\frac{1}{16},....$

Answer 1

We first use the first few terms to find the common ratio
$r=\frac{a_2}{a_1}=\frac{\left(\frac{1}{4}\right)}{(-\frac{1}{2})}=-\frac{1}{2}$
$r=\frac{a_3}{a_2}=\frac{(-\frac{1}{8})}{\left(\frac{1}{4}\right)}=-\frac{1}{2}$

$r=\frac{a_4}{a_3}=\frac{\left(\frac{1}{16}\right)}{(-\frac{1}{8})}=-\frac{1}{2}$

The common ration $r=-1/2$. We now use the formula $a_n=a_1\ast r^{n-1}$ for the $nth$ term to find $a_{20}$ as follows.
$a_{20}=a_1\ast r^{20-1}$
$\Rightarrow (-\frac{1}{2})\ast (-\frac{1}{2}{)}^{20-1}=\frac{1}{\left(2^{20}\right)}$