We all know what is the formula of the summation of ‘n’
natural numbers? Ask even a kid and he will promptly reply: n(n+1)/2
But have we ever wondered how the formula is drawn? How the
formula did come into existence?
In our school days (if taught), we were taught the summation
method to deduce the formula of the summation of ‘n’ natural numbers.
ΣK^2 = Σ(K+1)^2 – (n+1)^2; summation from k=0
to n. If we expand this equation, we will get the required formula; but this
not how the original formula was deduced.
When the great genius, Gauss was a child in the era of
1780’s, he found the formula. His teacher gave the class a sum to find the sum
of first 100 natural numbers. The exercise was given to engage the class for
some time. But the genius of Gauss was at work and he solved the problem in no
time. He found an interesting phenomenon.
Say the numbers are:
1 + 2 + 3 + 4 + …………………………………………………..+ 99 +100
100 + 99 + 98 +…………………………………………………. + 2 + 1
= 101, 101, 101 …………………………………………………… (100 times)
He observed that if the numbers are arranged in the reverse
order and then if the series are added, each of the numbers is the same i.e. we
get 100 couplets of 101 each. Hence the sum of the series is 101 * 100/2 =
5050.
Moreover, the phenomenon is the same for any given ‘n’
numbers.
Thus, if you were add the first ‘n’ natural numbers; the sum
would be ‘n’ times (n+1) divided by 2. And thus the great formula was found by
Sir Karl Friedrich Gauss in 1780’s.
Isn’t Mathematics beautiful, sure it is J
There exist a lot of patterns in the numbers, all it takes
is a keen eye to observe them and decipher the beauty.
Mathematics is truly the language of making the invisible –
Visible.
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