Don't understand the explanation of "sum of the factors In 5!"

[amusingminnow]

Why does 2^0+2^1+2^2+2^3=2^4-1 ?

[quacker88]

@amusingminnow Great question! This is a pattern that works for all powers of two, actually.

Let's add them one at a time and see if there's anything you notice:
$2^0=\boxed{1}$
$2^0+2^1=\boxed{3}$
$2^0+2^1+2^2=\boxed{7}$
$2^0+2^1+2^2+2^3=\boxed{15}$
$2^0+2^1+2^2+2^3+2^4=\boxed{31}$
What's special about all of the answers is that they're all $1$ less than the next power of $2$!

Knowing this is super helpful, it saves us from having to do a ton of addition. For example, we can use this shortcut when doing $2^0+2^1+2^2+...+2^9$, because we know that this is equal to the next power of $2,$ which is $2^{10}=1024$, minus $1$!

So $2^0+2^1+2^2+...+2^9=1023$.

If you really want to get into the nitty gritty of why this works, you have to factor. Let's use $2^4-1$.
Remember difference of squares? We can use that here!
$2^4-1=(2^2-1)(2^2+1)$
And since $2^2-1$ is a difference of two squares also, we can keep going.
$2^4-1=(2^2-1)(2^2+1)=(2-1)(2+1)(2^2+1)$.
Well, $(2-1)$ is just $1$, so the expression is equal to $(2+1)(2^2+1)$. Expand $(2+1)(2^2+1)$, and you'll get $2^3+2^2+2+1=2^3+2^2+2^1+2^0$.

So,
$2^3+2^2+2^1+2^0=2^4-1$.
You can use factoring to prove this for all of the powers of $2$