How to Add All the Numbers 1-100 In Under a Minute

Here’s a fun little math trick with a cool story behind it.  Ask the kids if they can figure out the sum of all the whole numbers between 1 and 100.  Then tell them about a kid who was able to figure it out in just a few seconds.

Carl Friedrich Gauss (1777-1855) is sometimes referred to as the “Prince of Mathematicians.”

https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Carl_Friedrich_Gauss_1840_by_Jensen.jpg/220px-Carl_Friedrich_Gauss_1840_by_Jensen.jpg

He was also a child prodigy.

When Carl was 9 or 10 years old in Germany, his teacher gave the class the assignment to add all the whole numbers from 1 to 100 (1+2+3+4+5 and so on) in order to keep them busy. Carl did the problem in his head almost immediately.  He wrote the answer on his slate, turned it in and sat down with his hands folded on his lap while (it is said) he teacher looked at him scornfully. When the teacher checked the stack of slates later, Carl was the only one to have the correct answer: 5050.

How did he do it?

Carl realized that he could add the numbers in pairs: 1+100, 2+99, 3+98, etc. and that every one of them added up to 101.

How to add all the numbers 1-100 in less than a minuteHe then simply multiplied 50 (the number of pairs of numbers he was adding) times 101 (the answer to every pair he added) to get 5050.

Here’s how to express this in algebra:

Use the formula Sn=n/2(a+l)

Sn is the sum of the numbers of terms

n is the total number of terms

a is the first term

l is the last term

Try it to figure out the sum of the numbers 1-10:

n= 10, a=1, l=10

So, Sn = 10/2(1+10)

= 5×11=55

Pretty cool, huh?