Chapter 12.3, Problem 8AYU

To determine

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

Answer to Problem 8AYU

The sequence is geometric with common ratio $r=-5$ and first four terms are . $-5,25,-125$ and $625$ ..

Explanation of Solution

Given information:

The given relation is.

  $\left\{s_n\right\}=\left\{{\left(-5\right)}^n\right\}$ .

Calculation:

Consider the given relation.

  $\left\{s_n\right\}=\left\{{\left(-5\right)}^n\right\}$ .

Calculate first four term of the given sequence.

  $\begin{array}{c}{s}_1=-5^1\\ =-5\\ s_2=-5^2\\ =25\end{array}$ .

  $\begin{array}{c}{s}_3=-5^3\\ =-125\\ s_4=-5^4\\ =625\end{array}$

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.

  $\begin{array}{c}r=\frac{25}{-5}=\frac{-125}{25}\\ r=-5=-5\end{array}$ .

Therefore, the sequence is geometric with common ratio $r=-5$ and first four terms are . $-5,25,-125$ and $625$ .