How would you find the sum of the first 50 odd numbers?

Answer 1

The answer is 2,500.

This is an interesting question as it turns out there are a few approaches we could take to solve this. After much thought I have come up with the following, which, if you are interested, you might want to consider:

The most boring way would be to sit there with a calculator and add them all up. Let's try something else.

You could realise that if there are 50 odd numbers we can split them in to 25 pairs which, here, each add up to 100 (e.g. 99 & 1, 97 & 3 etc... all the way to 51 & 49). Therefore the sum must be 25 * 100 = 2,500.

Another way, for this particular question, would be to realise that the sum of x number of consecutive odd digits (only when starting at 1 of course) is equal to x2. So where x = 50 the sum of the numbers is 502 = 2,500.

But why should that be? It is because the average value of the odd numbers will equal the first odd number plus the last odd number divided by 2. So for the first 50 odd numbers this would be 1 + 99 / 2 = 100 / 2 =50. So, in this scenario, the average will always equal the number of odd numbers! So the total must be this number squared.

Finally, we could try to come up with a formula to solve this, but I do not want to write too much. If you are interested in a formula then please message my board.