Chapter 12, Problem 19RE

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 19RE

The series is arithmetic, common difference is $4$ and the sum of first $n$ term is $2n\left(n-1\right)$ .

Explanation of Solution

Given: $0,4,8,12,\mathrm{....}$

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

  $\begin{array}{c}{b}_2-b_1=b_3-b_2\\ 4-0=8-4\\ 4=4\end{array}$

So, the common difference is $4$ .

The sum of first $n$ terms is calculated as,

  $\begin{array}{c}{S}_n=\frac{n}{2}\left(2a_1+\left(n-1\right)d\right)\\ =\frac{n}{2}\left(2\cdot 0+\left(n-1\right)4\right)\\ =n\left(2n-2\right)\\ =2n\left(n-1\right)\end{array}$

Therefore, the series is arithmetic, common difference is $4$ and the sum of first $n$ term is $2n\left(n-1\right)$