Chapter 12.3, Problem 12AYU

To determine

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

Answer to Problem 12AYU

The sequence is geometric with common ratio $r=3$ and first four terms are . $\frac{1}{3},1,3$ and $9$

Explanation of Solution

Given information:

The given relation is.

  $\left\{d_n\right\}=\left\{\frac{3^n}{9}\right\}$ .

Calculation:

Consider the given relation.

  $\left\{d_n\right\}=\left\{\frac{3^n}{9}\right\}$ .

Calculate first four term of the given sequence.

  $\begin{array}{c}{d}_1=\frac{3^1}{9}\\ =\frac{1}{3}\\ d_2=\frac{3^2}{9}\\ =1\end{array}$ .

  $\begin{array}{c}{d}_3=\frac{3^3}{9}\\ =3\\ d_4=\frac{3^4}{9}\\ =9\end{array}$

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

  $r=\frac{d_2}{d_1}=\frac{d_3}{d_2}$

Here $r$ is common ratio.

Put the values and simplify.

  $\begin{array}{c}r=\frac{1}{\frac{1}{3}}=\frac{3}{1}\\ r=3=3\end{array}$ .

Therefore, the sequence is geometric with common ratio $r=3$ and first four terms are . $\frac{1}{3},1,3$ and $9$ .