Question

Find the sum of each arithmetic series.

$\sum _{n=1}^7(2n+1)$

Short answer

The sum of the arithmetic series is63.

Step-by-step solution

Step 1. Given Information.

Given arithmetic series is $\sum _{n=1}^7(2n+1)$.

Step 2. Calculation.

From the summation, the number of terms is $n=7$

The first term is given by plugging $n=1$:

$\begin{array}{l}{a}_n=2n+1\\ a_1=2\left(1\right)+1\\ a_1=2+1\\ a_1=3\end{array}$

The last term is given by plugging $n=7$:

$\begin{array}{l}{a}_n=2n+1\\ a_7=2\left(7\right)+1\\ a_7=14+1\\ a_7=15\end{array}$

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, $a_1=3,n=7,a_n=15$

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(a_1+a_n)\\ S_7=\frac{7}{2}(3+15)\\ S_7=\frac{7}{2}\left(18\right)\\ S_7=7\left(9\right)\\ S_7=63\end{array}$

Step 3. Conclusion.

Hence, the sum is 63.