# How do you find the sum of the arithmetic sequence given 8, 3, -2, -7, -12, ....., -57?

Jul 9, 2016

Step 1: Find the number of terms

The nth term of an arithmetic sequence is given by ${t}_{n} = a + \left(n - 1\right) d$. We know ${t}_{n}$ (-57), r (-5) and a (8), however we don't know n. We will therefore solve for $n$.

$- 57 = 8 + \left(n - 1\right) - 5$

$- 57 = 8 - 5 n + 5$

$- 57 - 13 = - 5 n$

$- 70 = - 5 n$

$n = 14$

$\therefore$ The sequence has $14$ terms.

Step 2: Apply the formula for sum:

When dealing with arithmetic series, we will frequently use two formulas to help us find the sum. They are:

${s}_{n} = \frac{n}{2} \left(2 a + \left(n - 1\right) d\right)$

${s}_{n} = \frac{n}{2} \left(a + {t}_{n}\right)$

Since we know the first term, the last term and the number of terms, we have enough information to use the latter; this one is more efficient and easy to use than the first.

${s}_{n} = \frac{n}{2} \left(a + {t}_{n}\right)$

${s}_{14} = \frac{14}{2} \left(8 + \left(- 57\right)\right)$

${s}_{14} = 7 \left(8 - 57\right)$

${s}_{14} = - 343$

Hence, the sum of this arithmetic sequence is $- 343$.

Practice exercises:

1. Determine the sum of the following sequence:

$3 , 14 , 25 , \ldots , 179$

Hopefully this helps, and good luck!