Question

Find the indicated term of each geometric sequence.

$7th$term of$1,\frac{1}{2},\frac{1}{4},......$

Short answer

The$7th$term is$a_7=\frac{1}{64}$.

Step-by-step solution

Step 1. Given information.  

We have, $1,\frac{1}{2},\frac{1}{4},.......$

$a_1=1,a_2=\frac{1}{2},a_3=\frac{1}{4}$

The common ratio is

$$\begin{gathered} \frac{a_2}{a_1}=\frac{{\displaystyle \frac{1}{2}}}{1}=\frac{1}{2} \\ \frac{a_3}{a_2}=\frac{{\displaystyle \frac{1}{4}}}{{\displaystyle \frac{1}{2}}}=\frac{1}{2} \end{gathered}$$

Thus, the common ratio $r=\frac{1}{2}$

To find the 7th term.

The $nth$term of a geometric sequence is determined by formula $a_n=a_1{r}^{n-1}$

Substitute $a_1=1,r=\frac{1}{2}$and $n=7$.

$$\begin{gathered} a_7=1{\left(\frac{1}{2}\right)}^{7-1} \\ {a}_7={\left(\frac{1}{2}\right)}^6 \\ {a}_7=\frac{1}{64} \end{gathered}$$