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The Gauss Sumation Trick

One of the most famous mathematicians of all times was named Karl Gauss. One day, as the story goes, his teacher gave the class an assignment to keep them busy so he could take a nap in the back of the class. The problem he assigned would keep most of us busy for at least a half an hour, if not more. However, to his teacher's surprise, young Mr. Gauss solved it in seconds.

Here is the problem the teacher assigned. Students were told to add all the whole numbers from one to one hundred. That is, 1+2+3+4+5 …98+99+100. In less time than it took most students to write out this one hundred number addition problem, Gauss got the answer. The sum is 5,050 he told his teacher confidently, and so it was. But how did he arrive at this answer in so short a time?

Gauss was a genius, and geniuses sometimes see things differently than most of us non genius types. But that doesn't mean that we can't solve a problem like a genius would, having first been shown the way. Here is how young Gauss arrived at his answer so quickly. He observed that in the series of numbers:

1 +2 + 3 +4 …97 + 98+ 99 + 100,

the sum of pairs of numbers from each end, and working in toward the middle summed to the same value, 101. In other words, 1 + 100, 2 +99, 3 + 98, 4 + 97 etc. all sum to 101! Since there are fifty pairs of numbers in the series 1 to 100, Gauss reasoned that the sum of all the numbers would be 50 times 101 or 5,050.

Now that we know this little rule we can look for a pattern in the sum of the numbers in the following sets of series:
1 to 10 the sum is 55
1 to 100 the sum is 5050
1 to 1000 the sum is 500500
1 to 10000 the sum is 50005000
1 to 100000 the sum is 5000050000
1 to 1000000 the sum is 500000500000

Do you see the pattern?
Here is a hint. Notice that the sum is always 5 followed by one less the number of zeros in the highest number in the series, then another 5 and the same number of zeros again.

This little trick works fine when there is an even number of pairs, but what if the number of pairs is not even or you are just not sure?
There is a method that works any series, but you are going to need your calculator, pencil and paper.

Step #1 Add the first and last number in the series and write it down.
Step #2 Subtract the first number in the series from the last, add one to it, and then write it down.
Step #3 Multiply the value you got for step #1 times the value for step #2.
Step #4 Divide your answer by two and you have found the sum.

Let's use these rules to add the numbers in the series: 3,4,5,6,7 .

Rule #1 add first and last: 7+3 and we get 10

Rule #2 subtract the first from the last and then add 1: 7-3 = 4 and adding 1 we get 5.

Rule #3 multiply the result of rule #1 times the result of rule #2 That would be 10 times 5 or 50.

Rule #4 divide the result of rule #3 by 2. 50 divided by 2 is 25. So the sum must be 25. Prove it for yourself. It works!

Copyright 2021, William Johnson