Chapter 9.3, Problem 31E

To determine

To write: an expression for the $n$ th term of the geometric sequence and find the ${40}^{th}$ term.

Answer to Problem 31E

The expression for nth term is $a_n=500{\left(1.02\right)}^{n-1}$ .

The ${40}^{th}$ term of sequence is $a_{40}=1082.372384$ .

Explanation of Solution

Given information:

The given first term and common ratio,

  $a_1=500,r=1.02$

Calculation:

Consider the first time $a_1=500$ and common ration $r=1.02$

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 $a_1=500$ and $r=1.02$ as shown below.

  $\begin{array}{c}{a}_n=\left(500\right){\left(1.02\right)}^{n-1}\\ a_n=500{\left(1.02\right)}^{n-1}\end{array}$

Therefore, the formula for general term of geometric sequence is $a_n=500{\left(1.02\right)}^{n-1}$ .

Substitute $n=40$ into the formula $a_n=500{\left(1.02\right)}^{n-1}$ and simplify as shown below.

  $\begin{array}{c}{a}_{40}=500{\left(1.02\right)}^{n-1}\\ =500{\left(1.02\right)}^{40-1}\\ =500{\left(1.02\right)}^{39}\\ =1082.372384\end{array}$

Therefore, the ${40}^{th}$ term of sequence is $a_{40}=1082.372384$ .