What's 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + .... 65536

What's 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + .... 65536?

So previously we found how to find the sum of an arithmetic series (check that post out if you don't know what it means)
http://alexdoesphysics.blogspot.com/2018/04/whats-1-2-3-4-5-6-7-1000-solve-in-less.html?m=1 and we derived the formula. So what about a geometric series. This is a series where the next term is found by multiplying the previous number by a fixed number.

Lets define

a as the first term

r as the common ratio between each term

n as the number of terms in the sequence

So we can use this formula to find out what the answer to the original question.

We know that

a = 1

r = 2

n = 17 (if you know your powers of 2 well) 

(2 to the power of 16 is 65536 meaning there are 17 terms in the series)

And so plugging this into the formula gives 131071

Using this formula we can find the sum of any geometric series

e.g

5 + 10 + 20 + 40 + ... + 640

10 + 100 + 1000 + 10000 + 100000 + 1000000 + ... + 10000000000000000