To celebrate The Math Dude’s 100th episode, we’re going to find out why you should think of math as a puzzle and not a problem, and why math is actually fun, useful, and not at all nerdy.
Math Brain Game
To get started on our problem—or rather I should say puzzle—of investigating the sequence made by summing up positive odd integers, we should first make sure we know what the puzzle actually means. So, what are positive odd integers? Well, those are just the integers in the sequence 1, 3, 5, 7, 9, and so on (just start with 1 and keep adding 2 to find the next one). What happens when we add up these positive odd integers? Well, let’s start by looking at the sequence of numbers we get when we add up the first 1 positive odd integer, then the first 2 positive odd integers, then the first 3, and so on. The sequence looks like this:

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25
Of course, you could keep on doing this to your heart’s content and find more and more sums in the sequence. For example, the next one is the sum of the first 6 positive odd integers: 1 + 3 + 5 + 7 + 9 + 11 = 36; the one after that is the sum of the first 7 positive odd integers: 1 + 3 + 5 + 7 + 9 + 13 = 49; and on and on forever if you’d like (although I’m pretty sure you’d run out of time eventually).
To recap what we’ve got so far: we’ve found that when we add up increasing numbers of positive odd integers, we get the sequence of numbers: 1, 4, 9, 16, 25, 36, 49, and on and on. Do you see anything interesting about that sequence? Is there a pattern? If you do see something, that’s awesome! If not, don’t worry. Just like learning to solve crossword, Sudoku, or KenKen puzzles, it takes a while to develop an eye for this sort of thing. Instead of giving away the answer right now, I’m going to let you think about it for a while. And once you do find the pattern, the even bigger and more important question to think about is “Why does this happen?” That’s the real puzzle here!
What Is Math?
Before wrapping up, I’d like to ask what I think is a very important question about math that is rarely (if ever) asked. And that question is: What is math? As we’ve started to see today, and as we’ll continue to see next week when we finish solving our puzzle, my answer is that math should not be a boring and painful problem, but instead really should be a fun game for your brain. Of course, math is powerful, practical, and all of those good things too. But we shouldn’t lose sight of the fact that at their heart, math problems are really just puzzles for us to play around with.
Wrap Up
Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebookwhere you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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