### Instruction: The sum of a 15-term arithmetic series with first term 7  is $-210$.  What is the common difference?### Response: To find the common difference of an arithmetic series, I need to use the formula for the sum of an arithmetic series.
The formula is $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the $n$th term.
In this problem, I know that $S_n = -210$, $n = 15$, and $a_1 = 7$.
I can plug these values into the formula and simplify.
I get $-210 = \frac{15}{2}(7 + a_{15})$.
Multiplying both sides by 2, I get $-420 = 15(7 + a_{15})$.
Expanding the parentheses, I get $-420 = 105 + 15a_{15}$.
Subtracting 105 from both sides, I get $-525 = 15a_{15}$.
Dividing both sides by 15, I get $a_{15} = -35$.
Now I have the first term and the last term of the series.
To find the common difference, I can use the formula $a_n = a_1 + (n-1)d$, where $d$ is the common difference.
Plugging in $n = 15$, $a_1 = 7$, and $a_n = -35$, I get $-35 = 7 + (15-1)d$.
Simplifying, I get $-35 = 7 + 14d$.
Subtracting 7 from both sides, I get $-42 = 14d$.
Dividing both sides by 14, I get $d = -3$.
This is the common difference of the series.