Answer
$S_{15} = 840$
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:
$a_{1} = 7$
$a_{n} = 105$
We don't know what term the last number in the series, $105$, is, so we plug $105$ into the explicit formula for an arithmetic sequence, which will give us the number of the term $105$ 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 one term in the sequence from the one directly following it:
$d = 14 - 7 = 7$
The common difference $d$ is $7$.
Let's plug in the values we know into the explicit formula for an arithmetic sequence:
$105 = 7 + (n - 1)7$
Use distributive property:
$105 = 7 + 7n - 7$
Combine like factors on the right side of the equation:
$105 = 0 + 7n$
Divide each side of the equation by $7$:
$n = 15$
This means that there are $15$ 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:
$S_{15} = \frac{15}{2}(7 + 105)$
Evaluate what's in parentheses first:
$S_{15} = \frac{15}{2}(112)$
Multiply to solve:
$S_{15} = 840$
