Chapter 8, Problem 11CT

To determine

Find the recursive rule for the given sequence and the value of $a_9$ .

Answer to Problem 11CT

The recursive equation for the geometric series is $a_n=\left(-4\right)a_{n-1}$ and the explicit rule for the given geometric sequence is $a_n=5{\left(-4\right)}^{n-1}$ .

Explanation of Solution

Given:

The given sequence is $5,-20,80,-320,1280,---$ .

Calculation:

In a geometric sequence, the ratio of any term to the previous term, called the common ratio, is constant.

Find the common ratio of any term to the previous term of the sequence $5,-20,80,-320,1280,---$ .

  $\begin{array}{l}\frac{-20}{5}=-4\\ \frac{80}{-20}=-4\\ \frac{-320}{80}=-4\\ \frac{1280}{-320}=-4\end{array}$

The common ratio $\left(-4\right)$ is constant.

The recursive equation for the geometric series is $a_n=r{a}_{n-1}$ .

  $r=-4$

  $a_n=\left(-4\right)a_{n-1}$

The explicit rule for the geometric series is $a_n=a_1{r}^{n-1}$ .

Substitute the value of $a_1=5$ and the common ratio $r=-4$ in the explicit rule.

  $a_n=5{\left(-4\right)}^{n-1}$

Hence the recursive equation for the arithmetic series is $a_n=\left(-4\right)a_{n-1}$ and the explicit rule for the given geometric sequence is $a_n=5{\left(-4\right)}^{n-1}$ .