How to sum an arithmetic progression?

Whilst it is very easy to just plug the numbers into the formula you are given on your formula sheet there is a much faster and more intuitive method of doing this.


If we have n numbers beginning at a with an increment d then we can write our sequence as follows:


a, a+d, a+2d, a+3d,...., a+(n-1)d

If we wish to sum these, we can note that all of the terms pair up either side of the middle term giving pairs equal to 2a+(n-1)d.

We know that we must have n/2 of these pairs as we can only pair up values either side of the middle value (if it exists, which it does exclusively for odd n).

Thus we must just multiply our average term by the number of terms:

n/2 * (2a + (n-1) d)

Whilst this is just the formula you are given, if you know what it is doing then that enables you to take shortcuts, such as - if you are given the first and last term then you can simply replace 2a + (n-1)d with a+L.

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