Chapter 7.8, Problem 38PPE

To determine

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

Answer to Problem 38PPE

It is a geometric sequence. The recursive formula is $a_n=a_{n-1}.\left(2\right);a_1=5$

and explicit formula is $a_n=5.{\left(2\right)}^{n-1}$

Explanation of Solution

Given information : The given sequence is $5,10,20,40,\mathrm{...}$

We have the given sequence is $5,10,20,40,\mathrm{...}$

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 $a_1=5$ and common ratio = $\frac{10}{5}=2$

Now, $a_2=a_{1}r=5\times 2=10$

  $a_3=a_{2}r=10\times 2=20$ and so on

Therefore, the given sequence is a geometric sequence.

Now, we have

The recursive formula is

  $a_n=a_{n-1}.r;a_1=5$

Now, substituting the value of r

  $a_n=a_{n-1}.\left(2\right);a_1=5$

The explicit formula is

  $a_n=a_1{r}^{n-1}$

Substituting the value of a and r

  $a_n=5.{\left(2\right)}^{n-1}$