Chapter 8.2, Problem 53E

To determine

The sum of the finite arithmetic sequence.

Answer to Problem 53E

The sum of the given finite arithmetic sequence is $S_n=110$

Explanation of Solution

Given information:

  $2+4+6+8+10+12+14+16+18+20$

Formula used:

  $\frac{n}{2}\left(a_1+a_n\right)$

Calculation:

To calculate the sum: $2+4+6+8+10+12+14+16+18+20$

To begin, notice that the sequence is arithmetic with a common difference of 2.

Moreover, the sequence has 10 terms. So, the sum of the sequence is given by

  $\begin{array}{l}{S}_n=2+4+6+8+10+12+14+16+18+20\text{Given sum}\\ \text{=}\frac{n}{2}\left(a_1+a_n\right)\text{Formula to find the sum of a finite arithmetic sequence}\\ =\frac{10}{2}\left(2+20\right)\text{Substitute,}\{\begin{array}{c}n=10\\ a_1=2\\ a_{10}=20\end{array}\\ =\frac{10}{2}\left(22\right)\text{Add, the terms in parentheses}\end{array}$

Continue Simplification

  $\begin{array}{l}=\frac{10}{2}\times 22\text{Divide out the common factors}\\ =5\times 22\text{Clear out fractions}\\ =110\text{Multiply the terms}\end{array}$

The sum of the given finite arithmetic sequence is $S_n=110$

Conclusion:

The sum of the given finite arithmetic sequence is $S_n=110$