Answer
$S_{7} = 56$
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:
$a_{1} = 5$
$a_{n} = 11$
We don't know what term the last number in the series, $11$, is, so we plug $11$ into the explicit formula for an arithmetic sequence, which will give us the number of the term $11$ 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:
$d = 6 - 5 = 1$
The common difference $d$ is $1$.
Let's plug in the values we know into the explicit formula for an arithmetic sequence:
$11 = 5 + (n - 1)1$
Use distributive property:
$11 = 5 + n - 1$
Combine like factors on the right side of the equation:
$11 = 4 + n$
Subtract $4$ from each side of the equation:
$n = 7$
This means that there are $7$ 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:
$S_{7} = \frac{7}{2}(5 + 11)$
Evaluate what's in parentheses first:
$S_{7} = \frac{7}{2}(16)$
Multiply to solve:
$S_{7} = 56$