Chapter 7.8, Problem 24PPE

To determine

Tofind:The explicit formula for thegiven geometric sequence.

Answer to Problem 24PPE

The explicit formula for the geometric sequence $200,40,8,1\frac{3}{5},\dots$ is $a_n=200\cdot {\left(\frac{1}{5}\right)}^{n-1}$ .

Explanation of Solution

Given information:

The given geometric sequence is $200,40,8,1\frac{3}{5},\dots$ .

Conceptused:

A geometric sequence is a sequence in which the ratio consecutive terms is constant.

A sequence of the form $a_1,a_{1}r,a_1{r}^2,a_1{r}^3,\dots ,a_1{r}^{n-1},\dots$ is the geometric with $a_1$ as the first term, r the common ratio, and nth term $a_n=a_1{r}^{n-1}$ . This is also known as explicit formula for the sequence.

The geometric sequence may be obtained as $\langle s\rangle =a_1,a_2,a_3,a_4,\dots$

The given geometric sequence is $\langle s\rangle =200,40,8,1\frac{3}{5},\dots$ , that may be written as

  $\langle s\rangle =200,200\times \left(\frac{1}{5}\right),200\times {\left(\frac{1}{5}\right)}^2,200\times {\left(\frac{1}{5}\right)}^3,\dots ,200\times {\left(\frac{1}{5}\right)}^{n-1},\dots$ .

The first term of the sequence is 200 and the common ratio is $\frac{1}{5}$ .

Therefore, the explicit formula for the sequence is $a_n=200\cdot {\left(\frac{1}{5}\right)}^{n-1}$ .

Conclusion:

The explicit formula for the geometric sequence $\langle s\rangle =200,40,8,1\frac{3}{5},\dots$ is $a_n=200\cdot {\left(\frac{1}{5}\right)}^{n-1}$ .