Ask Uncle Colin: about summation notation

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I’m having trouble getting my head around sum notation! I can’t tell whether $\sum_{n=0}^{6}{5}$ means $0\times5 + 1\times 5 + … + 6\times 5$ or $0 + 1 + 2 + … + 6$ or $5 + 5 + 5 … + 5$. Wolfram|Alpha just gives me the answer, not an explanation!

Seems I’m Generating Mistakes Again

Dear SIGMA,

Picasso1 said of computers, “They are useless. They can only give you answers.” Obviously, our friend Pablo was being provocative and wrong, but in this instance, he has a bit of a point: Wolfram|Alpha spitting out the answer “35” doesn’t really help you here (unless you’re prepared to work out all of the options and see which of them makes 35… but even then, that’s an example rather than an explanation.)

So, in the interests of being better than a computer, here’s how you break down the sigma notation.

  • The squiggle, as you’ve correctly identified, means “add a list of stuff up”. Your difficulty seems to be working out precisely what you need to add up
  • The numbers above and below the squiggle tell you where to start and stop counting in the list. Here, you start counting at 0 and end at 6 (meaning you’ll be adding up seven things).
  • The stuff to the right of the squiggle tells you what’s in the list — here, every element of the list is ‘5’

So, in this case, the first term ($n=0$) is 5. The second term ($n=1$) is also 5. So is the third, and every term up to the 7th term ($n=6$), which — you’ve guessed it — is 5. Your sum is $5 + 5 + 5 + 5 + 5 + 5 + 5 = 7 \times 5 = 35$.

Now, had the thing to the right of the sum been $n$ rather than 5, you’d go through the same steps: the first term (when $n=0$) is 0; the second term ($n=1$) is 1, and so on — the sum would be $0 + 1 + 2 + 3 + 4 + 5 + 6 = 21$.

Similarly, if the thing you needed to add up was $5n$, you’d work it out for each term. When $n=0$, you get 0. When $n=1$, you get 5. When $n=2$, you get 10, and so on up to $n=6$, when you get 30. The sum would be $0 + 5 + 10 + … + 30 = 105$.

Sigma notation is very compact and can be confusing at first — knowing when to work things out term by term and when to use a formula (for arithmetic or geometric series, for example) is something that comes with experience and the frustration of “oh… there was an easier way?!”.

I hope that helps!

— Uncle Colin

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. Who would probably have preferred to sum cubics, I suppose []

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