## 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!