To determine the number of terms [tex]\( n \)[/tex] in the arithmetic series, we need to use the given values:
- The first term [tex]\( a_1 = -3 \)[/tex]
- The last term [tex]\( a_n = 44 \)[/tex]
- The sum of the first [tex]\( n \)[/tex] terms [tex]\( S_n = 328 \)[/tex]
The formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic series is:
[tex]\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\][/tex]
Given [tex]\( S_n = 328 \)[/tex], [tex]\( a_1 = -3 \)[/tex], and [tex]\( a_n = 44 \)[/tex], we can plug these values into the formula and solve for [tex]\( n \)[/tex].
First, substitute the known values into the formula:
[tex]\[
328 = \frac{n}{2} \cdot (-3 + 44)
\][/tex]
Calculate the sum inside the parentheses:
[tex]\[
-3 + 44 = 41
\][/tex]
Thus, the equation becomes:
[tex]\[
328 = \frac{n}{2} \cdot 41
\][/tex]
To isolate [tex]\( n \)[/tex], multiply both sides of the equation by 2:
[tex]\[
656 = n \cdot 41
\][/tex]
Finally, divide both sides by 41:
[tex]\[
n = \frac{656}{41}
\][/tex]
Simplify the fraction:
[tex]\[
n = 16
\][/tex]
Therefore, the number of terms in the arithmetic series is:
[tex]\[
n = 16
\][/tex]
