Chapter 12.2, Problem 39AYU

In Problems 39-56, find each sum.

$7+12+17+\cdots +\left(2+5n\right)$

To determine

To find: The sum of the given sequence.

Answer to Problem 39AYU

$\frac{n}{2}\left[9+5n\right]$

Explanation of Solution

Given:

$7+12+17+\dots +\left(2+5n\right)$

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 $\left\{a_n\right\}=\left\{2+5n\right\}$ is an arithmetic sequence with 1^st term $a_1=7$ and $n$^th term.

$a_n\text{ = 2 + 5}n$

To find the sum $S_n$:

$S_n={\displaystyle \sum }_{k=1}^n\left(2+5k\right)=\frac{n}{2}\left[7+2+5n\right]=\frac{n}{2}\left[9+5n\right]$