Chapter 7.8, Problem 41PPE

To determine

To Write: Write the explicit and recursive formula if given sequence is a geometric sequence.

Answer to Problem 41PPE

It is a geometric sequence. The recursive formula is $a_n=a_{n-1}.\left(\frac{1}{7}\right);a_1=98$

and explicit formula is $a_n=98.{\left(\frac{1}{7}\right)}^{n-1}$

Explanation of Solution

Given information : The given sequence is $98,14,2,\frac{2}{7},\mathrm{...}$

We have the given sequence is $98,14,2,\frac{2}{7},\mathrm{...}$

A sequence is said to be geometric sequence if each term is obtained by multiplying the common ratio to the previous term.

Here the first term is $a_1=98$ and common ratio = $\frac{14}{98}=\frac{1}{7}$

Now, $a_2=a_{1}r=98\times \frac{1}{7}=14$

  $a_3=a_{2}r=14\times \frac{1}{7}=2$ and so on

Therefore, the given sequence is a geometric sequence.

Now, we have

The recursive formula is

  $a_n=a_{n-1}.r;a_1=98$

Now, substituting the value of r

  $a_n=a_{n-1}.\left(\frac{1}{7}\right);a_1=98$

The explicit formula is

  $a_n=a_1{r}^{n-1}$

Substituting the value of a and r

  $a_n=98.{\left(\frac{1}{7}\right)}^{n-1}$