Answer
$S_{8} = 92$
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 $nth$ last term.
List the values we are given:
$a_{1} = 8$
$a_{n} = 15$
We don't know what term the last number in the series, $15$, is, so we plug $15$ into the explicit formula for an arithmetic sequence, which will give us the number of the term $15$ in the series:
$a_{n} = a_{1} + (n - 1)d$, where $a_{n}$ is a certain term in a sequence, $a_{1}$ is the first term in the sequence, $n$ is the number of the term within the sequence, and $d$ is the common difference.
The common difference $d$ is found by subtracting a term from the one directly following it:
$d = 9 - 8 = 1$
The common difference $d$ is $1$.
Let's plug in the values we know into the explicit formula for an arithmetic sequence:
$15 = 8 + (n - 1)1$
Use distributive property:
$15 = 8 + n - 1$
Combine like factors on the right side of the equation:
$15 = 7 + n$
Subtract $7$ from each side of the equation:
$n = 8$
This means that there are $8$ terms in this finite series.
Let's plug in the values that we are given into the formula to find the sum of a finite arithmetic series:
$S_{8} = \frac{8}{2}(8 + 15)$
Simplify the fraction:
$S_{8} = 4(8 + 15)$
Evaluate what's in parentheses first:
$S_{8} = 4(23)$
Multiply to solve:
$S_{8} = 92$
