How do I add up the integers from 1 to 1000 without going insane?

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Suppose you are asked by your teacher to sum all the integers from 1 up to 1000. You might be thinking they must hold some kind of grudge against you. If you try to calculate the sum by adding on each integer one at a time, you will have to perform 999 separate additions, some of which will be quite long and tedious. This would take far longer than anyone can be bothered to spend adding up numbers.

There is, however, a quicker way. Let's give our sum a letter to represent its unknown value. Let's use "S" for "sum". Then:

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

But we can rewrite this sum in reverse order. Starting with 1000 and ending with 1:

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

We can now add together these two equations to give us:

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

Simplifying both sides gives us:

2S = 1001 + 1001 + 1001 + ... + 1001 + 1001

The right hand side has 1000 separate terms, since our original sum contained 1000 numbers. So:

2S = 1000 x 1001 = 1001000

Dividing both sides by two we find that S = 500500. Therefore:

1 + 2 + 3 + ... + 999 + 1000 = 500500

This calculation is an example of a more general concept called an arithmetic series, where you sum a sequence of numbers which differ by adding on a fixed amount with each step.

Matthew B. A Level Maths tutor, A Level Further Mathematics  tutor, A...

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