Question

Find the sum of each arithmetic series.

$5+11+17+\mathrm{...}+95$

Short answer

The sum of the arithmetic series is800.

Step-by-step solution

Step 1. Given Information.

Given arithmetic series is$5+11+17+\mathrm{...}+95$ .

Step 2. Calculation.

The difference between the terms is$11-5=6$

The nth term of an arithmetic series is given by $a_n=a_1+(n-1)d$

Here, $a_1=5,d=6,a_n=95$

Plugging the values:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ 95=5+(n-1)\left(6\right)\\ (n-1)\left(6\right)=95-5\\ (n-1)\left(6\right)=90\\ n-1=15\\ n=15+1\\ n=16\end{array}$

The sum $S_n$ of the first n terms of an arithmetic series is given by

$S_n=\frac{n}{2}(a_1+a_n)$

Here, $a_1=5,n=16,a_n=95$

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(a_1+a_n)\\ S_{16}=\frac{16}{2}(5+95)\\ S_{16}=8\left(100\right)\\ S_{16}=800\end{array}$

Step 3. Conclusion.

Hence, the sum is 800.