# How do you find the sum of the arithmetic series 2 + 5 + 8 + ... + 56?

Mar 31, 2016

${s}_{n} = 551$

#### Explanation:

The sum of an arithmetic series is ${s}_{n} = \frac{n}{2} \left({a}_{1} + {a}_{n}\right)$
Where n is the number of terms
a_1 is the first term and
a_n is the last tern or the nth term

but in the given problem the nth term is not given , but we can determine its term by using the formula ${a}_{n} = {a}_{1} + \left(n - 1\right) d$
Where d is the common difference

Given : $2 + 5 + 8 + \ldots + 56$
${a}_{n} = 56$
${a}_{1} = 2$
$d = 3$( just by substracring the second and the first term since we already know this is an arithmetic sequence, it has common difference)
n=?
By substituting to the equation
$56 = 2 + \left(n - 1\right) 3$
$54 = 3 n - 3$
$57 = 3 n$
$n = 19$

So we already have the number of terms.
Now solving for the sum of the arithmetic sequence using the formula ${s}_{n} = \frac{n}{2} \left({a}_{1} + {a}_{n}\right)$

using the givens we have , substitute to the sum formula
${s}_{n} = \frac{19}{2} \left(2 + 56\right)$
${s}_{n} = \frac{19}{2} \left(58\right)$
${s}_{n} = 19 \left(29\right)$
${s}_{n} = 551$