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.
1 + 2 + 3 + … + 98 + 99 + 100
101
101
101
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.
©2007 http://mathmomblog.wordpress.com
This worksheet maybe copied for classroom use.
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:
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©2007 http://mathmomblog.wordpress.com
This worksheet maybe copied for classroom use.
2 + 4 + 6 + … + 36 + 38 + 40
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____
11 + 12 + 13 + … + 48 + 49 + 50
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____
5 + 6 + 7 + … + 63 + 64 + 65
How many numbers:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____
1 + 4 + 7 + … + 64 + 67 + 70
How many numbers*:_____
How many pairs:_____
Sum of each pair:_____
Overall sum:_____
*Careful, this one’s tricky. See if you can figure this out by finding
a pattern and a rule. Explain what you did below:
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©2007 http://mathmomblog.wordpress.com
This worksheet maybe copied for classroom use.
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:________
©2007 http://mathmomblog.wordpress.com
This worksheet maybe copied for classroom use.