Chapter 12, Problem 15RE

To determine

To find: Whether the series is arithmetic, geometric or neither, if the series is arithmetic find the common difference and sum of first $n$ terms and if the series is geometric find the common ratio and the sum of first $n$ terms.

Answer to Problem 15RE

The sequence is neither arithmetic nor geometric.

Explanation of Solution

Given: $\left\{c_n\right\}=\left\{2n^3\right\}$

The terms of the series are,

  $\begin{array}{c}{c}_1=2{\left(1\right)}^3=2\\ c_2=2{\left(2\right)}^3=16\\ c_3=2{\left(3\right)}^3=54\\ c_4=2{\left(4\right)}^3=128\end{array}$

If the series is arithmetic then it must the follow the relation,.

  $\begin{array}{c}{c}_2-c_1=c_3-c_2\\ 16-2=54-16\\ 14\ne 38\end{array}$

The sequence is not arithmetic.

Foe the sequence to be geometric it must follow,

  $\begin{array}{c}\frac{c_2}{c_1}=\frac{c_3}{c_2}\\ \frac{16}{2}=\frac{54}{16}\\ 8\ne \frac{27}{8}\end{array}$

So, the sequence is not geometric.

Therefore, the sequence is neither arithmetic nor geometric.