Chapter 7.8, Problem 51PPE

To determine

To find:A geometric sequence and the explicit and recursive formulas for the sequence.

Answer to Problem 51PPE

The sequence is $9,27,81,243,\mathrm{...}$

The recursive formula is $a_1=9;a_n=a_{n-1}\cdot 3$

The explicit formula is $a_n=9\cdot 3^{n-1}$

Explanation of Solution

Given information:The sequence is geometric.

Calculation:Let the geometric sequence be $9,27,81,243,\mathrm{...}$

The first term $a_1$ is 9 and the common ratio r is $\frac{27}{9}=3$

The recursive formula is given by,

  $\begin{array}{l}{a}_1=a;a_n=a_{n-1}\cdot r\\ \Rightarrow a_1=9;a_n=a_{n-1}\cdot r[\text{Substituting the first term }{a}_1\text{=9]}\\ \Rightarrow a_1=9;a_n=a_{n-1}\cdot 3[\text{Substituting the common ratio }r=3]\end{array}$

Thus, the recursive formula is $a_1=9;a_n=a_{n-1}\cdot 3$

The explicit formula is given by,

  $\begin{array}{l}{a}_n=a_1\cdot r^{n-1}\\ \Rightarrow a_n=9\cdot r^{n-1}[\text{Substituting the first term }{a}_1\text{=9]}\\ \Rightarrow a_n=9\cdot 3^{n-1}[\text{Substituting the common ratio }r=3]\end{array}$

Thus, the explicit formula is $a_n=9\cdot 3^{n-1}$ .