Chapter 12.3, Problem 11AYU

To determine

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

Answer to Problem 11AYU

The sequence is geometric with common ratio $r=2$ and first four terms are . $\frac{1}{4},\frac{1}{2},1$ and $2$ .

Explanation of Solution

Given information:

The given relation is.

  $\left\{c_n\right\}=\left\{\frac{2^{n-1}}{4}\right\}$ .

Calculation:

Consider the given relation.

  $\left\{c_n\right\}=\left\{\frac{2^{n-1}}{4}\right\}$ .

Calculate first four term of the given sequence.

  $\begin{array}{c}{c}_1=\frac{2^{1-1}}{4}\\ =\frac{1}{4}\\ c_2=\frac{2^{2-1}}{4}\\ =\frac{1}{2}\end{array}$ .

  $\begin{array}{c}{c}_3=\frac{2^{3-1}}{4}\\ =1\\ c_4=\frac{2^{4-1}}{4}\\ =2\end{array}$

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

  $r=\frac{c_2}{c_1}=\frac{c_3}{c_2}$

Here $r$ is common ratio.

Put the values and simplify.

  $\begin{array}{c}r=\frac{\frac{1}{2}}{\frac{1}{4}}=\frac{1}{\frac{1}{2}}\\ r=2=2\end{array}$ .

Therefore, the sequence is geometric with common ratio $r=2$ and first four terms are . $\frac{1}{4},\frac{1}{2},1$ and $2$ .