There are things I'm supposed to teach and they're strung along into a curriculum and I often get to some of these things and I'm like, "wtf? why?" So when I get to those things I think, okay, why might this ever be useful or interesting.
Why might someone want to sum an arithmetic sequence? That's a list of numbers that keep increasing or decreasing by the same amount. So for example, 3, 5, 7, 9, ... is an arithmetic sequence and so is 8, 3, -2, -7, -12, -17, ...
The only times this has ever been useful in my life is when I needed a shortcut for adding a bunch of things really fast. Which doesn't come up that much, but it does come up once in a blue moon.
So I decided to impress the children with my lightning addition capability. "Impress" is maybe a tad ambitious as words go because they are 16 but their interest can be piqued.
When they walked in I gave them a slip of paper with 20 boxes and asked them to come up with an arithmetic sequence and write down the first 20 terms. They could use their favorite numbers, or their least favorite numbers, or numbers they were indifferent toward. I made a little form special for this purpose because if one wants humans to take one seriously, one must make it obvious that one has prepared for their arrival. (And plus I'm already best friends with the guy in my department with the good paper cutter.)
The little form also had a spot for the sum of the 20 terms. They were to add them up on a calculator and write it down.
Then I asked them to trade slips with a partner and check each other's math before handing it in, because if my answers didn't match what was on their paper, there was going to be hell to pay.
They wrote their names on the list of 20 things. They cut off the sum and handed me the list of twenty things. They kept the sum. The anticipation was building. And by that I mean I said, "Children. Prepare to be amazed," and the children made me try again because I was too monotone.
So then I shuffled up the little slips of sequences and started saying, B, your sum is 210. C, your sum is 384. D, your sum is 2440. E, your sum is -24.
They were astonished! I only made two mental arithmetic errors in two class periods, which was convincing enough that they wanted to know my secret. We wrote some sequences on the board and we stared at them for a while. No one figured it out. Then I was the worst magician in the world and spilled the trick. For educational purposes.
Did they learn anything from this little stunt? No. But they were ready to learn something, which is saying something.