Page 4 - Matematica_Mathematics - Suma lui Gauss_Gauss sum

3. Gauss sum

Karl Friedrich Gauss lived from 1777 to 1855
and was a great German mathematician, physicist
and astronomer. He is considered one of the greatest
German scientists.
It is said that in primary classes he was punished
by his teacher to calculate the sum of the first 100
natural numbers. He noticed that he could use a
calculation technique that allowed him to solve the
exercise in seconds, to his teacher’s surprise. Therefore,
Gauss’s method comes to help us and not to make our
work difficult.
But let’s see how he thought about solving this
task!
He wrote twice this sum, placing the numbers in
the string first ascending, then descending and noted
the sum by S because he did not know its value.

1 + 2 + 3 + 4 + 5 + ......... + 98 + 99 + 100 = S

100 + 99 + 98 + 97 + 96 + ......... + 3 + 2 + 1 = S

101 + 101+ 101+ 101 + 101+...........+101+ 101+ 101 = S + S (i.e. 2 × S)

He then noticed that by adding the numbers written one below the other in the two strings, he obtained
every time 101. Adding 100 times to 101, he got the double of the sum of the numbers from 1 to 100.
From here things became very simple.

101 × 100 = 2 × S

10100 = 2 × S, and to find out the sum, he divided the product by 2.

Therefore, 1 + 2 + 3 + 4 + 5 + ......... + 98 + 99 + 100 = 100 × 101 : 2

So, to find out the sum of the numbers
from 1 to 100 we multiply the last number (100)
by its successor (101) then divide by 2.
The good part is that we can apply this
formula to find out the sum of the numbers
in a string, no matter how many they are.
The condition is that the string must be built
according to a certain rule.