Answer
$S_5 = 20$
Work Step by Step
To find the sum of a finite series, we can use the formula:
$S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the series, $n$ is the number of terms in the series, $a_1$ is the first term, and $a_n$ is the $nth$ term.
We already have $a_1$ and $n$; however, we need to find $a_n$. This can be done by using the explicit formula for arithmetic sequences, which is given by $a_n = a + (n - 1)d$. We can find $d$ by subtracting one term from the following term:
$d = 7 - 10 = -3$
Plug values into the explicit formula:
$a_5 = 10 + (5 - 1)(-3)$
Evaluate expressions within parentheses first:
$a_5 = 10 + (4)(-3)$
Multiply:
$a_5 = 10 + (-12)$
Add to solve:
$a_5 = -2$
Now we can plug the values that are known into the formula to find the sum of a finite arithmetic series:
$S_5 = \frac{5}{2}(10 + (-2))$
Evaluate what is in parentheses first:
$S_5 = \frac{5}{2}(8)$
Multiply:
$S_5 = \frac{40}{2}$
Simplify the fraction:
$S_5 = 20$
