Question

Find $S_n$for each arithmetic series described.

$d=5,n=16,a_n=72$

Short answer

$S_n$for the arithmetic series is$S_{16}=552$ .

Step-by-step solution

Step 1. Given Information.

Given arithmetic series is $d=5,n=16,a_n=72$.

Step 2. Calculation.

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

Here, $d=5,n=16,a_n=72$

Plugging the values:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ 72=a_1+(16-1)\left(5\right)\\ 72=a_1+\left(15\right)\left(5\right)\\ 72=a_1+75\\ a_1=72-75\\ a_1=-3\end{array}$

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

$S_n=\frac{n}{2}(2a_1+(n-1)d)$

Here, $a_1=-3,n=16,d=5$

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(2a_1+(n-1)d)\\ S_{16}=\frac{16}{2}(2(-3)+(16-1)\left(5\right))\\ S_{16}=8(-6+\left(15\right)\left(5\right))\\ S_{16}=8(-6+75)\\ S_{16}=8\left(69\right)\\ S_{16}=552\end{array}$

Step 3. Conclusion.

Hence, the sum is $S_{16}=552$$S_{16}=552$.