Chapter 12.3, Problem 14AYU

To determine

To find: Prove the sequence geometric and find common ratio along with first four terms.

Answer to Problem 14AYU

The sequence is geometric with common ratio $r=9$ and first four terms are . $9,81,729$ and $6561$ .

Explanation of Solution

Given information:

The given relation is.

  $\left\{f_n\right\}=\left\{3^{2n}\right\}$ .

Calculation:

Consider the given relation.

  $\left\{f_n\right\}=\left\{3^{2n}\right\}$ .

Calculate first four term of the given sequence.

  $\begin{array}{c}{f}_1=3^{2\times 1}\\ =9\\ f_2=3^{2\times 2}\\ =81\end{array}$ .

  $\begin{array}{c}{f}_3=3^{2\times 3}\\ =729\\ f_4=3^{2\times 4}\\ =6561\end{array}$

For the sequence to be geometric this relation must be true.

  $r=\frac{f_2}{f_1}=\frac{f_3}{f_2}$

Here $r$ is common ratio.

Put the values and simplify.

  $\begin{array}{c}r=\frac{81}{9}=\frac{729}{81}\\ r=9=9\end{array}$ .

Therefore, the sequence is geometric with common ratio $r=9$ and first four terms are . $9,81,729$ and $6561$ .