Chapter 12.2, Problem 38AYU

In Problems 39-56, find each sum.

$2+4+6+\cdots +2n$

To determine

To find: The sum of the given sequence.

Answer to Problem 38AYU

$n\left(n+1\right)$

Explanation of Solution

Given:

$2+4+6+\mathrm{...}+2n$

Formula used:

Sum of the first $n$ terms of an arithmetic sequence.

$S_n=\frac{n}{2}\left[a_1+a_n\right]$

Calculation:

The sequence $\left\{a_n\right\}=\left\{2n\right\}$ is an arithmetic sequence with $1^{st}$ term $a_1=2$ and $n^{th}$ term $a_n=2n\mathrm{}$.

The sum $S_n$:

$S_n={\displaystyle \sum _{k=1}^n\left(2k\right)}=\frac{n}{2}\left[2+2n\right]=n\left(n+1\right)$