How To Add Up All The Numbers From 1 To 1000

I had a friend who was asked at a job interview, years ago for an internship, to add up all the numbers from 1 to 1000. It was meant as a brainteaser, a test of his intelligence. He didn`t find it very hard at all, because he knew that there was a trick. There was no need to add up the numbers one at a time.

So, what was it that he did?

He knew that he had to find the sum.

S = 1 + 2 +3 + ... 999 + 1000

That`s a thousand numbers.

We can write the sum backwards.

S = 1000 + 999 + 998 + ... 2 + 1

Then we can add these two expressions together.

2S = (1 + 1000) + (2 + 999) + (3 + 998) + ... (999 + 2) + (1000 + 1)

we have a thousand terms, each of which adds up to 1001

2S = 1000 x 1001

So, our sum S = 1000 x 1001 / 2

That`s easy 500 times 1001 = 500,500.

So, my friend answered the question and he got the job. lt;/p>

Now, do you see how the formula works.

Number of terms = 1000

Average term = 1/2 (first term plus last term)

So, the answer is number of terms times the average term.

We can do the same for 1 + 2 + 3 + .... 99 + 100

That`s 100 numbers and the average term is 50.5.

So, 100 times 50.5 equals 5050.

This works whenever we have a series where the terms go up by the same amount each time .

We call that an arithmetic progression. The formula only works for a series like that.

Once again - the sum of an arithmetic progression is the number of terms times the average term.

That`s how my friend knew how to answer the question.