I don't often see a lot of pure maths posts here, so I thought I would contribute, with one of the most useful formulas I use in everyday life (well kind of)

The other day, someone asked me what is the average roll on a 6 sided die (for those that aren't familiar with the term, a die is the singular form of dice)

Their immediate guess was 3? but that's not actually right, the average is 3.5.

So how would we calculate this, simple add up all the outcomes and divided by the number of outcomes.

1+2+3+4+5+6 = 21, so we divided that by 6, and obtain 3.5.

So what's a quick way to add up a series

well a cube die, only has 6 sides so that's nice and easy to add up, but what if it was the one of those die with 20 sides. that will take a bit longer. Luckily there is an easy way to ass up a series. In this case the series is called an arithmetic series.

Arithmetic series

An arithmetic series is a sum of numbers, where the next number is the same distance away from the previous number by the same amount

examples include a series that adds 1 each time

1+2+3+4+5..

or a series of even numbers

2+4+6+8

the numbers can start from any number like this

10+13+16+19

How to solve these quickly

a very simple trick, is you add two series together say for example you had the series 1 to 6 going up by ones, we do the following

(1+ 2+ 3+ 4 +5 +6 ) + (6+5+4+3+2+1)

This becomes (7+7+7+7+7+7) or simply 7*6 = 42

now we divide by 2, and have the answer which is 21. If you remember the dice example we would simply divide by 6 to get 3.5 as the answer.

An even quicker way

well the above process there is a formula that works, I wont derive the formula here, but it is derived in much the same way the problem was solved above. The formula is:

Sum = (a+l)*n/2

where
a = first number
l = last number
n = how many numbers

in our example, Sum = (1 + 6)* 6/2 = 7*3 = 21

So there you have it an easy way to calculate sums, perhaps you might need it one day.