Chapter 11, Problem 19CLR

To determine

To calculate:

The sequence is arithmetic or geometric or neither and $n^{th}$ term of $108,-72,48,-32,\mathrm{...}$

Answer to Problem 19CLR

The sequence is geometric and $n^{th}$ term is $108\cdot {\left(-0.6\right)}^{n-1}$ .

Explanation of Solution

Given information:

  $108,-72,48,-32,\mathrm{...}$

First term: $a=108$

Formula used:

  $a_n=a_1\cdot r^{n-1}$

Calculation:

Consider the sequence:

  $108,-72,48,-32,\mathrm{...}$

The common difference is:

  $\begin{array}{l}d=a_2-a_1=\left(-72\right)-108=-180\\ d=a_3-a_2=48-\left(-72\right)=120\\ d=a_4-a_3=\left(-32\right)-48=-80\end{array}$

As per result, this sequence is not arithmetic sequence.

The common ratio is:

  $\begin{array}{l}r=\frac{a_2}{a_1}\\ r=\frac{-72}{108}=\frac{48}{-72}=\frac{-32}{48}=--0.6\end{array}$

As per result, this sequence is geometric sequence.

The formula for $n^{th}$ term:

  $\begin{array}{l}{a}_n=a_1\cdot r^{n-1}\\ a_n=108\cdot {\left(-0.6\right)}^{n-1}\end{array}$

The sequence is geometric and $n^{th}$ term is $108\cdot {\left(-0.6\right)}^{n-1}$ .