Sum of a series - Part 1

So we know the general formula for a sum of an arithmetic series is

$\displaystyle \sum_{a}^{n} = \frac {n(n + 1)}{2}$

Where $n$ is the number of terms in the series and $a$ is the starting number

The problem with this formula though is that it only handles when $a=1$

So what if we want the formula when $a \neq 1$

Let's say that for now $a = 3$ and $n = 6$ so we have an answer to check our equation with.

So

$\displaystyle \sum_{3}^{6} = 3 + 4 + 5 + 6 + 7 + 8 = 33$

This sum is almost the same as

$\displaystyle \sum_{1}^{8} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36$

The second sum can be worked out using the formula

$\displaystyle \sum_{1}^{8} = \frac {8(8 + 1)}{2} = 36$

And the first one is just

$\displaystyle \sum_{3}^{6} = \displaystyle \sum_{1}^{8} - (1 + 2)$

Since $(1 + 2)$ can be represented as $\displaystyle \sum_{1}^{2}$, we can represent our equation as

$\displaystyle \sum_{3}^{6} = \displaystyle \sum_{1}^{8} - \displaystyle \sum_{1}^{2}$

And since both of those can be represented using the formula we can rewrite it as

$\displaystyle \sum_{3}^{6} = \frac {8(8 + 1)}{2} - \frac {2(2 + 1)}{2} = 36 - 3 = 33$

Rewriting this using $a$ and $n$ gives us

$\displaystyle \sum_{a}^{n} = \displaystyle \sum_{1}^{n + (a - 1)} - \displaystyle \sum_{1}^{a - 1}$

$\displaystyle \sum_{a}^{n} = \frac {(n + (a - 1))((n + (a - 1)) + 1)}{2} - \frac {(a - 1)((a - 1) + 1)}{2}$

$\displaystyle \sum_{a}^{n} = \frac {(n + a)(n + a - 1)}{2} - \frac {a^2 - a}{2}$

Lets test it on our example

$\displaystyle \sum_{3}^{6} = \frac {(6 + 3)(6 + 3 - 1)}{2} - \frac {3^2 - 3}{2} = 36 - 3 = 33$

So it works (at least for $a = 3$ and $n = 6$), feel free to test it for yourself.