More fun with counting and geometry

We finished section 3.3 of Art of Problem Solving’s Introduction to Counting and Probability today. One of the problems in the section asked about a few relatively basic integer sums. It made for a fun little morning project.

The first sum I asked the boys to look at was 1 + 2 + 3 + \ldots + n. They’d seen this sum already in the examples from the book, but I wanted to hear them talk through it on their own:

The next sum was the first n positive even integers 2 + 4 + 6 + \ldots + 2n. The tiny bit of extra difficulty for kids here is keeping track of what n is. Sorry about the garbage truck stopping by in the middle of this video . . . .

The next sum was the first n positive odd integers. There are a couple of ways to approach this sum, and the answer is a bit of a surprise:

Since we just found an amazingly simple formula for the sum off the first positive n odd integers, we brought out the snap cubes to see if we could find the connection to geometry:

So, a fun little project. A few basic sums that we encounter when learning to count give us some nice practice with both arithmetic and basic algebra, and we also get a surprising and pretty cool connection to geometry. Nice morning!