# What is the sum of the arithmetic series 2 + 5 + 8 + ... + 53?

Feb 21, 2016

First, we must find the number of terms, n.

#### Explanation:

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

$53 = 2 + \left(n - 1\right) 3$

$53 = 2 + 3 n - 3$

$54 = 3 n$

$18 = n$

Now that we know the number of terms we can use the formula ${s}_{n} = \frac{n}{2} \left({t}_{1} + {t}_{n}\right)$

${s}_{18} = \frac{18}{2} \left(2 + 53\right)$

${s}_{18} = 9 \left(55\right)$

${s}_{18} = 495$

The sum is of 495.

When finding the sum of an arithmetic series, there are two formulas that you may use: the one presented above and ${s}_{n} = \frac{n}{2} \left\{2 a + \left(n - 1\right) d\right\}$. You use the latter when you don't know the last term.

Practice exercises:

1. Find the sum of the following series: $5 , 11 , 17 , \ldots , 131$

2. Find the sum of a series with the following characteristics.

3. First three terms: $7 , - 1 , - 9 , \ldots$

4. $n = 33$

Hopefully this helps!