Chapter 7.8, Problem 22P

Write the recursive formula for each geometric sequence.

$7,14,28,56,\dots$

To determine

To write a recursive rule for the geometric sequence $7,14,28,\mathrm{56......}$ .

Answer to Problem 22P

The recursive rule for given sequence is $a_1=7,a_n=\left(2\right)a_{n-1}\text{,}\text{where}{a}_{n-1}=7{\left(2\right)}^{n-2}.$

Explanation of Solution

Given information :

Given geometric sequence $7,14,28,\mathrm{56......}$ .

The given geometric sequence $7,14,28,\mathrm{56......}$ . has the first term $a_1=7$ and common ratio $r=\left(\frac{14}{7}\right)=2$ .

The $n^{th}$ term is $a_n=a_1{r}^{n-1}\Rightarrow a_n=7{\left(2\right)}^{n-1}\Rightarrow a_{n-1}=7{\left(2\right)}^{n-2}$

Thus, the recursive rule for a geometric sequence is $a_n=r.a_{n-1}$

  $\Rightarrow a_n=\left(2\right)a_{n-1}$

Hence, the recursive rule for given sequence is $a_1=7,a_n=\left(2\right)a_{n-1}$