The idea behind the sum of powers of 2 I know that the sum of power of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.
For example, sum of n numbers is $\frac{n(n+1)}{2}$. The idea is that we replicate the set and put it in a rectangle, hence we can do the trick. What is the logic behind the sum of power of $2$ formula?

Another geometric interpretation. 
Start with a line segment, AB. Extend this into a line. Use compassto mark off a point C by setting center at B and passing radius through A. Then lengths, AB=BC. Create point D in a similar way. We have thus constructed a length double that of the original line segment.
This doubling can continue AS MUCH S you like.
Now ignore the first line segment, AB.
The series of segments now starts with BD which has double the length of AB. 
In general the nth segment in the first case is the n-1th segment upon removal of AB. Further, each is double the length of its corresponding section. 
So By subtracting a part we've doubled what remains, but with fewer sections to account for.
This gives the formula.
