Answer
$S_{10} = -165$
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:
$a_{1} = -3$
$a_{n} = -30$
We don't know what term the last number in the series, $-30$, is, so we plug $-30$ into the explicit formula for an arithmetic sequence, which will give us the number of the term $-30$ 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:
$d = -6 - (-3) = -3$
The common difference $d$ is $-3$.
Let's plug in the values we know into the explicit formula for an arithmetic sequence:
$-30 = -3 + (n - 1)(-3)$
Use distributive property:
$-30 = -3 - 3n + 3$
Combine like factors on the right side of the equation:
$-30 = -3n$
Divide each side of the equation by $-3$:
$n = 10$
This means that there are $10$ 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:
$S_{10} = \frac{10}{2}(-3 + (-30))$
Simplify the fraction:
$S_{10} = 5(-3 + (-30))$
Evaluate what's in parentheses first:
$S_{10} = 5(-33)$
Multiply to solve:
$S_{10} = -165$
