Gauss' Method
Carl Friedrich Gauss, a mathematician in the 1700s, found a method of finding the sum of consecutive numbers. A sequence of numbers in which the same number is added to each term to get the next is an arithmetic sequence.
Take for example, the sum of the numbers from 1 to 100, like this:
- Write out the sums:
1 + 2 + 3 + 4 . . . + 97 + 98 + 99 + 100 ← there are 100 numbers
-
Cut the sums in half, turn the last half around and add them to the first half:
1 2 3 4 . . . 48 49 50
100 +99 +98 +97 + . . . + 53 + 52 + 51
101 + 101 + 101 + 101 + ... + 101 + 101 + 101 ← they all add to 101!
- There are 100/2 = 50 of these sums (because you cut them in half) so this is:
50 x 101 = 5050
How about the sum of the first 50 odd numbers? Well this is:
- 1 + 3 + 5 + 7 + . . . + 93 + 95 +97 + 99 ← this is 50 numbers
-
Cut the sums in half, turn the last half around and add them to the first half:
1 3 5 7 . . . 45 47 49
+99 +97 +95 +93 + . . . + 55 + 53 + 51
100 + 100 + 100 + 100 + . . . + 100 + 100 + 100
- There are 50/2 = 25 of these sums (because you cut them in half) x 100 = 25 x 100 = 2500
How about the sum of the first 50 even numbers? Well this is:
- 2 + 4 + 6 + 8 + . . . + 94 + 96 + 98 + 100 ← this is also 50 numbers!
-
Cut the sums in half, turn the last half around and add them to the first half:
2 4 6 8 . . . 46 48 50
+100 +98 +96 +94 + . . . + 56 + 54 + 52
102 102 102 102 + . . . 102 102 102
- There are 50/2 = 25 of these sums (because you cut them in half) x 102 = 25 x 102 = 2550
Notice that if you add the sum of the odd numbers and the sum of the even numbers you get the sum of all the numbers from 1 to 100: 2500 + 2550 = 5050.
This is as it should be!
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