Pairing up with Gauss

[A number of people seem to be finding this post by searching for things like "adding up numbers in a sequence" or "what is the sum of all the numbers from..." or "summing arithmetic sequences" etc. I hope that you will stay and read this article, even though it doesn't give you the formula. It walks through the method of finding such sums at a level appropriate for about 4th grade and up, and the derivation of the formula at a level appropriate for middle schoolers. Once you've learned to derive the formula, you need not memorize it, since you can re-derive it as needed.]

Here is of a lesson I developed for group of 10- to 12-year-olds. I plan to use it the first time I meet with them this year. I will first ask them to think about how they might add up all the numbers from 1 to 100, and we’ll talk about the different ways it can be done, then talk about the Gauss story, and play with lots of different kinds of sequences. I have shown many of them the Gauss trick before when playing with triangle numbers, etc. But I’m not sure how many will remember it, and in any case, this lesson is designed to take them to the next level.

They have had only very limited exposure to algebraic notation and variables, so I’m pushing things at the end, but we’ll see how it goes. I may spread this out over two class periods; I don’t think one will give them time to play with the numbers and try playing with the variables, but I’m never sure how long things will take, so again, we’ll see how it goes…

Here’s a nice illustration of some other ways of adding up sequences like this. (Full article here.) I’m hoping that my group will come up with some of these during our initial discussion of how we might approach adding up these numbers.

Young Gauss’s “trick” for finding the sum of an arithmetic progression is usually explained in terms of adding pairs of elements from opposite ends of the sequence, so that all the pairs have the same sum. One way to envision this process is to fold the series in half with a hairpin bend. Another approach is to write the series twice, once in ascending and once in descending order. A third method selects just a single pair of elements, typically the first and last, in order the calculate the average. Finally, some tellers of the story point out that the formula for summing the first n natural numbers also generates the nth triangular number; in effect, the sum is half the area of an n-by-n+1 rectangle. Brian Hayes

Adding Arithmetic Sequences by Pairing Off

Legend has it that when the great mathematician Carl Gauss was a young boy, his teacher asked him to add all the numbers from 1 to 100. Gauss quickly realized that there was a fast way of doing this, paired numbers from each end, and multiplied by the number of pairs.

We can see that this sequence contains several pairs, each of which adds up to 101.

Now all we need to do is figure out how many pairs there are. Since there are 100 numbers, and there are 2 numbers in each pair, there are 50 pairs. So there are 50 101’s to add up. When we add 101 50 times, we get 50 × 101 = 5050. So…

1 + 2 + 3 + … + 98 + 99 + 100 = 5050.

Pairing Up – Practice

Use the pairing up method to find each of the following sums:

1 + 2 + 3 + … + 28 + 29 + 30
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____

1 + 2 + 3 + … + 48 + 49 + 50
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____

1 + 2 + 3 + … + 23 + 24 + 25
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____
Use this sequence to explain why the same method even works for sequences containing odd numbers of entries.

(I’m going to cut out all of the “routine” questions for the rest here, but they’re all on the worksheet for the kids.)

2 + 4 + 6 + … + 36 + 38 + 40
11 + 12 + 13 + … + 48 + 49 + 50
5 + 6 + 7 + … + 63 + 64 + 65

1 + 4 + 7 + … + 64 + 67 + 70
How many numbers*:_____
*Careful, this one’s tricky. See if you can figure this out by finding a pattern and a rule. Explain what you did.

Pairing up – Finding the Rules

1 + 2 + 3 + … + (n-2) + (n-1) + n
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:________

2 + 4 + 6 + … + (2n-4) + (2n-2) + 2n
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:________

a + (a+d) + (a+2d) + … + (a+(n-3)d) + (a+(n-2)d) + (a+(n-1)d)
Fill in this description: This is a sequence of _____ numbers starting from _____ and increasing by _____ each time.
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:________

Here’s a link to a copy of my worksheet. Feel free to use it, as long as you retain my copyright notice at the bottom of each page. If you find it useful, I’d love it if you’d let me know.

Pairing Off With Gauss

Related Post: Visit To Guangyang Primary School (relates a 4th grade lesson on this topic — scroll way down)

Follow-ups:

Pairing Up with Gauss — Follow-up I 

Pairing Up with Gauss — Follow-up II

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