Question

Find the sum of each arithmetic series.

$38+35+32+\mathrm{...}+2$

Short answer

The sum of the arithmetic series is260.

Step-by-step solution

Step 1. Given Information.

Given arithmetic series is $38+35+32+\mathrm{...}+2$.

Step 2. Calculation.

The difference between the terms is $35-38=-3$

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

Here, localid="1648547769020" $a_1=38,d=-3,a_n=2$

Plugging the values:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ 2=38+(n-1)(-3)\\ (n-1)(-3)=2-38\\ (n-1)(-3)=-36\\ n-1=12\\ n=12+1\\ n=13\end{array}$

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, localid="1648547799867" $a_1=38,n=13,a_n=2$

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(a_1+a_n)\\ S_{13}=\frac{13}{2}(38+2)\\ S_{13}=\frac{13}{2}\left(40\right)\\ S_{13}=13\left(20\right)\\ S_{13}=260\end{array}$

Step 3. Conclusion.

Hence, the sum is 260.