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# Sum of a series - Part 2

On the previous note in this series we learnt / revised that

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

That's the formula when the difference ($$d$$) is $$1$$ so what would the formula be if $$d \neq 1$$

Let's denote the whole thing as $$\displaystyle \sum_{a}^{n} d$$

So let's say that $$a = 3$$, $$n = 6$$ and $$d = 2$$ what equation would we get from that.

$\displaystyle \sum_{3}^{6} 2 = 3 + 5 + 7 + 9 + 11 + 13 = 48$

We're going to have to use a different method to last time to solve this.

Since $$a = 3$$ and $$d = 2$$ we can put those in to get

$\displaystyle \sum_{a}^{6} d = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) + (a + 5d)$

That can be written as

$\displaystyle \sum_{a}^{6} d = 6a + (1 + 2 + 3 + 4 + 5)d$

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

This is basically a simplified version of the previous equation with a $$d$$ added in to account for the difference. This formula however is still flawed as it can only handle a constant variable for $$d$$.

Note by Jack Rawlin
2 years, 5 months ago