How to Add Up Powers of Numbers


Do you need to know the formula to tell you what the sum of the first N counting numbers, raised to a power? No, you do not. Not really. It can save a bit of time to know the sum of the numbers raised to the first power. Most mathematicians would know it, or be able to recreate it fast enough:

\sum_{n = 1}^{N} n = 1 + 2 + 3 + \cdots + N = \frac{1}{2}N\left(N + 1\right)

But there are similar formulas to add up, say, the counting numbers squared, or cubed, or so. And a toot on Mathstodon, the mathematics-themed instance of social network Mastodon, makes me aware of a cute paper about this. In it Dr Alessandro Mariani describes A simple mnemonic to compute sums of powers.

It’s a neat one. Mariani describes a way to use knowledge of the sum of numbers to the first power to generate a formula for the sum of squares. And then to use the sum of squares formula to generate the sum of cubes. The sum of cubes then lets you get the sub of fourth-powers. And so on. This takes a while to do if you’re interested in the sum of twentieth powers. But do you know how many times you’ll ever need to generate that formula? Anyway, as Mariani notes, this sort of thing is useful if you find yourself at a mathematics competition. Or some other event where you can’t just have the computer calculate this stuff.

Mariani’s process is a great one. Like many mnemonics it doesn’t make literal sense. It expects one to integrate and differentiate polynomials. Anyone likely to be interested in a formula for the sums of twelfth powers knows how to do those in their sleep. But they’re integrating and differentiating polynomials for which, in context, the integrals and derivatives don’t exist. Or at least don’t mean anything. That’s all right. If all you want is the right answer, it’s okay to get there by a wrong method. At least if you verify the answer is right, which the last section of Mariani’s paper does. So, give it a read if you’d like to see a neat mathematical trick to a maybe useful result.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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