Chapter 11.7, Problem 4MRE

To determine

To calculate:

Tell whether the sequence $-24,12,6$ is arithmetic, geometric or neither and find the formula for the nth term of the sequence.

Answer to Problem 4MRE

The given sequence $-24,12,6$ is a geometric sequence and the nth term of the sequence is $a_n=2^{-n+4}\cdot 3{\left(-1\right)}^{n-1}$ .

Explanation of Solution

Given information:

The given sequence is,

  $-24,12,6$

Formula:

The formula of geometric sequence is,

  $a_n=a_1{r}^{n-1}$

Where,

  $r$ : Common ratio value

  $a_n$ : Nth term

  $a_1$ : $1st$ Term of the geometric sequence

Calculation:

The given sequence is,

  $-24,12,6$

First check if the sequence is arithmetic,

Here, $\begin{array}{l}{a}_1=-24\\ a_2=12\\ a_3=6\end{array}$

Now, $\begin{array}{l}{a}_2-a_1=12-\left(-24\right)=36\\ a_3-a_2=6-12=6\end{array}$

As the difference between the two consecutive numbers is not same it is not an arithmetic series,

Then, check for Geometric sequence

By using the formula of geometric sequence,

  $a_n=a_1{r}^{n-1}$

Substituting the values in above formula,

Here, First term: $-24$

Second term: $12$

Third term: $6$

  $a_n=-\frac{24{\left(-1\right)}^{n-1}}{2^{n-1}}$

  $\begin{array}{l}{a}_n=-\frac{24{\left(-1\right)}^{n-1}}{2^{n-1}}=3\cdot 2^{-n+4_{\left(-1\right)}}{}^{n-1}\\ a_n=-\left(-\frac{24{\left(-1\right)}^{n-1}}{2^{n-1}}\right)\left[\text{Applying}\text{factorial}\text{rule:}\frac{-a}{b}=-\frac{a}{b}\right]\\ =-\left(-3\cdot 2^{-n+4}{\left(-1\right)}^{n-1}\right)\left[\text{Cancel}\frac{24{\left(-1\right)}^{n-1}}{2^{n-1}}:3\cdot 2^{-n+4}{\left(-1\right)}^{n-1}\right]\\ a_n=2^{-n+4}\cdot 3{\left(-1\right)}^{n-1}\end{array}$