Chapter 7.8, Problem 28P

Determine if each sequence is a geometric sequence. If it is, find the common ratio and write the explicit and recursive formulas.

$25,50,100,200,\dots$

To determine

To find: If the sequence is a geometric sequence. If it is, find the common ratio and write the explicit and recursive formulas.

Answer to Problem 28P

The given sequence is geometric with common ratio $r=2$

The explicit formula for the given sequence is $a_n=25{\left(2\right)}^{n-1}$

The recursive formula for the given sequence is $a_1=25,a_n=2a_{n-1}$

Explanation of Solution

Given sequence: $25,50,100,200,\mathrm{......}$

Calculation:

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed , non-zero number called common ratio.

Since, each term (except the fist term) is found by multiplying the previous term by 2, therefore, the given sequence is geometric and the common ratio of the geometric sequence is $r=2$ .

The first term of the sequence is $a_1=25$

The explicit formula for the given sequence is

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ a_n=25{\left(2\right)}^{n-1}\end{array}$

The recursive formula for the given sequence is

  $\begin{array}{l}{a}_1=25,a_n=a_{n-1}\cdot r\\ a_1=25,a_n=a_{n-1}\left(2\right)\\ a_1=25,a_n=2a_{n-1}\end{array}$