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A common math problem in school is to add the first 100 numbers. In other words find the answer to:
This problem is related counting the number of objects in a triangle shape as shown below. Because of this they are known as the 'Triangular Numbers'.
The problem of adding the first 100 numbers then is the same as finding the 100th triangular number. This still doesn't help us solve the problem. The question is, is there a pattern in this we could use? The answer is 'OF COURSE!'.
Let's start with adding just the first 2 numbers.
if we add the columns instead of going across we always get 3!
There are now two threes to add which makes six but since we added the number 1+2 twice we need to divide the 6 by 2 and get 3 for the sum. This seems like a lot of work to add the numbers 1 and 2 but works well as we want to add higher and higher numbers.
Lets look at what happens if we try and add the first 10 numbers.
Notice that now we get ten sums of 11. Multiplying 10x11=110 but again we added the numbers twice so we must divide this by 2 giving us a sum of 55 for the first 10 numbers.
If you notice in both cases we had the highest number we added times one more than that number and then we divided by 2.
If we did this for the first 30 numbers we would multiply 30x31=930 and then dividing this by 2 would give us 465 for the sum of the first 30 numbers.
That's all there is to it!
Bet you could easily add the first 1000 numbers now. Try it and then look below to see if you are right.
For further information try this great article.
The sum of the first 1000 numbers is 500,500. Hope you got it right.