Aha Moment, trapezoid and the series

Math is wonderful. Especially, when it all connects. Last spring I was teaching arithmetic series to my honors 8th grade class, and I made a connection that I had never made before. For those who don't know a arithmetic series is the sum of (generally) a lot of numbers that increase by steady amounts, such as 5+9+13+...101. The goal for our us is to find the sum with out actually doing all the dirty work of adding up all the numbers (easy for Gauss, but hard for many of us). There is a formula for this which is to add the first and the last (5 and 101 in my example), multiply by the number of numbers (25 in my case, harder to figure out than it sounds; the famous fence-post problem) and divide by 2. For those of you that look cool formulas it looks something like this:



The standard way of seeing this is with some pictures of bar graphs each bar representing one summand. The sum that we are trying to find is like adding up all the lengths represented by this bar graph



One nice way to see the formula is to take a copy of the graph and place it on top of the first but running backward from last to first as in this picture



Then each bar has the same height, the sum of the first and last, and we can multiply by the number of bars to find the total. We doubled what we are trying to find so to find the sum of the red columns I take that answer and divide by 2.

How do trapezoids fit in?

We if you look carefully, you will see the that the original picture is pretty much a trapezoid.



How do you find the area of this trapezoid? You take the length of the parallel sides, in this case 5 and 101. Multiply by the height between those sides, I guess this would be how many bars there are 25, and divide by 2. And in fact to proof that this formula works for trapezoids is exactly the same. Double the trapezoid to create a parallelogram that you already know how to find the area of.

So arithmetic series are isomorphic to trapezoids. Who knew?