Question

Find the sum of each arithmetic series.

$\sum _{k=3}^7(3k+4)$

Short answer

The sum of the arithmetic series is95.

Step-by-step solution

Step 1. Given Information.

Given arithmetic series is $\sum _{k=3}^7(3k+4)$.

Step 2. Calculation.

From the summation, the number of terms is $n=5$

The first term is given by plugging $k=3$:

$\begin{array}{l}{a}_k=3k+4\\ a_3=3\left(3\right)+4\\ a_3=9+4\\ a_3=13\end{array}$

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

$\begin{array}{l}{a}_k=3k+4\\ a_7=3\left(7\right)+4\\ a_7=21+4\\ a_7=25\end{array}$

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, $a_1=13,n=5,a_n=25$

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(a_1+a_n)\\ S_5=\frac{5}{2}(13+25)\\ S_5=\frac{5}{2}\left(38\right)\\ S_5=5\left(19\right)\\ S_5=95\end{array}$

Step 3. Conclusion.

Hence, the sum is 95.