Question

What is the sum of the finite arithmetic series?

Answer 1

The underlying arithmetic progression is

-15, 0, 15, 30, ...

starting with $a_1=-15$ and with a common difference of $d=15$ between terms. Recursively, this sequence is given by

$\begin{cases}a_1=-15\\a_n=a_{n-1}+15&\text{for }n>1\end{cases}$

So we have

$a_2=a_1+15$

$a_3=a_2+15=a_1+15\cdot2$

$a_4=a_3+15=a_1+15\cdot3$

and so on, so that the explicit rule for the sequence is

$a_n=a_1+15(n-1)=15n-30$

for $n\ge1$.

The series consists of 15 terms, since

$195=15n-30\implies15n=225\implies n=15$

So we have

$-15+0+15+\cdots+195=\displaystyle\sum_{n=1}^{15}(15n-30)=15\cdot\frac{15\cdot16}2-30\cdot15=1350$

and the answer would be B.