Gauss in His Childhood - A Glimpse of Talent

Talent is something you can do easily while others find it difficult to do.
Genius is something you can do easily while others find it impossible to do.

When Gauss (1777-1855) was a child, he attended the primary school in his local town. One day, the teacher found the class was too unsettled. In order to keep the class quiet for a while, he asked the children to sum the numbers from 1 to 100 before they could be dismissed. The class did stay silent while everyone was busy sketching their calculations on papers. Gauss didn't move. He stared at the blackboard for a few seconds, raised his hand and said, " The answer is 5050." The teacher looked at Gauss in disbelief, thinking he was just keen to leave the classroom. Gauss then explained step by step as follows:

"We call the sum of the numbers 1 to 100 S. Then,

S = 1 + 2 + .................. + 99 + 100

If we reverse the order of the numbers and add, the sum is still the same S.

S = 100 + 99 + .................. + 2 + 1

If we take the two together (i.e. add them vertically), we have two times the sum 2S, but the terms become 101 all the way for the 100 terms.

2S = 101 + 101 + .......... + 101 + 101

Since the numbers are the same, we don't need to add 100 times but we can multiply by 100. Simpler still, multiplying by 100 is just to add two zeros at the end of the number.

2S = 101 x 100 = 10100

To get back to the sum S, we just divide by 2.

S = 10100/2 = 5050."

Q.E.D.

Following on Gauss idea, we can develop further to generalize to sum the numbers from 1 to N (no matter what N is):

1 + 2 + ............... + N = N(N+1)/2

or, to sum from the number M to number N:

M + (M+1) + .......... + (N-1) + N = (N-M+1)(N+M)/2

If we still want to go further, we can research more to sum their squares:

1 + 2² + 3² + ………. + N²

or their cubes,

1 + 2³ + 3³ + ………. + N³

or, to push to the extreme,

1 + 2^m + 3^m + ………. + N^m

(no matter what m is).

But these require further mathematical techniques like recursive formula and Bernouilli numbers. I can touch on more as time permits.