Chapter 9.3, Problem 32E

To determine

To write: an expression for the $n$ th term of the geometric sequence and find the ${60}^{th}$ term.

Answer to Problem 32E

The expression for nth term is $a_n=1000{\left(1.005\right)}^{n-1}$ .

The ${60}^{th}$ term of sequence is $a_{60}=1342.139455$ .

Explanation of Solution

Given information:

The given first term and common ratio,

  $a_1=1000,r=1.005$

Calculation:

Consider the first time $a_1=1000$ and common ration $r=1.005$

The general term of geometric sequence is given by formula

  $a_n=a_1{r}^{n-1}$

Here, $a_1$ is the first term and $r$ is common ratio.

Substitute $a_1=1000$ and $r=1.005$ as shown below.

  $\begin{array}{c}{a}_n=\left(100\right){\left(1.005\right)}^{n-1}\\ a_n=1000{\left(1.005\right)}^{n-1}\end{array}$

Therefore, the formula for general term of geometric sequence is $a_n=1000{\left(1.005\right)}^{n-1}$ .

Substitute $n=60$ into the formula $a_n=1000{\left(1.005\right)}^{n-1}$ and simplify as shown below.

  $\begin{array}{c}{a}_{60}=1000{\left(1.005\right)}^{n-1}\\ =1000{\left(1.005\right)}^{60-1}\\ =1000{\left(1.005\right)}^{59}\\ =1342.139455\end{array}$

Therefore, the ${60}^{th}$ term of sequence is $a_{60}=1342.139455$ .