Chapter 12.3, Problem 74AYU

To determine

To find: Whether the sequence is arithmetic, geometric or neither, find the common difference or common ratio and also find the sum of the first fifty terms if the sequence is arithmetic or geometric.

Answer to Problem 74AYU

The sequence arithmetic and the common difference is $2$ and the sum of first $50$ terms is $2550$ .

Explanation of Solution

Given information:

The series is $2,4,6,\mathrm{8.....}$ .

Calculation:

Here, $a_1=2$ , $a_2=4$ and $a_3=6$ .

Calculate the difference between successive terms.

  $\begin{array}{c}{a}_2-a_1=4-2\\ =2\end{array}$

  $\begin{array}{c}{a}_3-a_2=6-4\\ =2\end{array}$

Calculate the last term.

  $l=2n$

Calculate the sum of first $50$ terms.

  $\begin{array}{c}{S}_n=\frac{n}{2}\left(a+l\right)\\ =\frac{n}{2}\left(a+2n\right)\\ =\frac{50}{2}\left(2+2\left(50\right)\right)\\ =2550\end{array}$

Therefore, the sequence arithmetic and the common difference is $2$ and the sum of first $50$ terms is $2550$ .