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The Great Gauss Summation 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
that 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 after being shown the way that
we can not 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 pair 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.

This little trick will work for any series of numbers provided that they are
evenly spaced. For example, 2, 4, 6, 8, or 3, 6, 9, 12, or, 5, 10, 15, 20 etc.
In every case, to find the sum you need only add the first and the last and
then multiply by the number of pairs in the series.

We can write the solution to all problems of this type using the following
mathematical sentence:

The sum of a series = (first + last) times the number of pairs in the series.

Let's try this out with the series 5 + 10 + 15 +20 + 25 +30

The sum of this series = (first + last) times the number of pairs

Or, the sum of this series =( 5 + 30) times 3 pairs
=(35) x 3

Check it out for yourself by adding the numbers the hard way, one at a time!

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, then another 5 and the same
number of zeros again.