Q9. - Algebra 2

Question

Find the sum of each arithmetic series.

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

Step-by-Step Solution

Answer

The sum of the arithmetic series is 260.

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,  $a_1=38, d=-3, a_n=2$

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,  $a_1=38, n=13, a_n=2$

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.