Chapter 12.3, Problem 7AYU

To determine

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

Answer to Problem 7AYU

The sequence is geometric with common ratio $r=3$ and first four terms are . $3,9,27$ and $81$ .

Explanation of Solution

Given information:

The given relation is.

  $\left\{s_n\right\}=\left\{3^n\right\}$ .

Calculation:

Consider the given relation.

  $\left\{s_n\right\}=\left\{3^n\right\}$ .

Calculate first four term of the given sequence.

  $\begin{array}{c}{s}_1=3^1\\ =3\\ s_2=3^2\\ =9\end{array}$ .

  $\begin{array}{c}{s}_3=3^3\\ =27\\ s_4=3^4\\ =81\end{array}$

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

  $r=\frac{s_2}{s_1}=\frac{s_3}{s_2}$

Here $r$ is common ratio.

Put the values and simplify.

  $\begin{array}{c}r=\frac{9}{3}=\frac{27}{9}\\ r=3=3\end{array}$ .

Therefore, the sequence is geometric with common ratio $r=3$ and first four terms are . $3,9,27$ and $81$ .