Dear Uncle Colin,

I noticed the sum of an arithmetic series and the formula for the area of a trapezium were very similar. How are they related?

- Watching Every Interesting Equation, Recognised Something Trapeziumesque Regarding Arithmetic Sequence Sums


Indeed, they are related, as you can see from this horizontal sketch of a simple arithmetic sequence:

*** 3 **** +4 ***** +5 ****** +6 ******* +7

Lookit: a trapezium! The parallel sides have lengths of 3 and 7, the height is 5 rows, so its area is 25 - exactly how many stars there are.

That can be generalised to an area of $\frac{n}{2}(a+l)$.

There is a small wrinkle, in that (on close inspection) the slanted ‘top’ of the trapezium doesn’t exactly match up with the top of each bar, but the discrepancies turn out to cancel out.

In fact, the standard proofs for the area formula and for the series sum are almost identical: copy the shape (or series), turn it around, and add it back on to make a rectangle (or constant series). Work out the area (or sum) of the doubled shape (series) and halve the result.

Hope that helps!

- Uncle Colin