Question

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

• A. ${20}^{20}$
• B. $\frac{1}{{20}^{20}}$
• C. $\frac{1}{20}$
• D. $\frac{1}{{20}^{19}}$

Answer 1

The correct option is B $\frac{1}{{20}^{20}}$

We first use the first few terms to find the common ratio.
$r=\frac{a_2}{a_1}=\frac{\frac{1}{4}}{-\frac{1}{2}}=-\frac{1}{2}$
$r=\frac{a_3}{a_2}=-\frac{\frac{1}{8}}{\frac{1}{4}}=-\frac{1}{2}$
$r=\frac{a_4}{a_3}=\frac{\frac{1}{16}}{-\frac{1}{8}}=-\frac{1}{2}$
The common ratio $r=-\frac{1}{2}$. We now use the formula $a_n=a_1\times r^{n-1}$ for the $n^{th}$ term to find $a_{20}$ as follows.
$a_{20}=a_1\times r^{20-1}$
$=(-\frac{1}{2})\times {(-\frac{1}{2})}^{20-1}$
$=\frac{1}{{20}^{20}}$