Answer
$S_{11} = 176$
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:
$a_{1} = 1$
$a_{n} = 31$
We don't know what term the last number in the series, $31$, is, so we plug $31$ into the explicit formula for an arithmetic sequence, which will give us the number of the term $31$ 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:
$d = 4 - 1 = 3$
The common difference $d$ is $3$.
Let's plug in the values we know into the explicit formula for an arithmetic sequence:
$31 = 1 + (n - 1)3$
Use distributive property:
$31 = 1 + 3n - 3$
Combine like factors on the right side of the equation:
$31 = -2 + 3n$
Add $2$ to each side of the equation:
$33 = 3n$
Divide each side of the equation by $3$:
$n = 11$
This means that there are $11$ 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:
$S_{11} = \frac{11}{2}(1 + 31)$
Evaluate what's in parentheses first:
$S_{11} = \frac{11}{2}(32)$
Multiply to solve:
$S_{11} = 176$