Prove that for all positive integers $n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$

Question

I want to prove that for all positive integers $n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$. By mathematical induction:

1) it holds for $n=1$, since $2^1=2^2-2=4-2=2$

2) if $2+2^2+2^3+...+2^n=2^{n+1}-2$, then prove that $2+2^2+2^3+...+2^n+2^{n+1}=2^{n+2}-2$ holds.

I have no idea how to proceed with step 2), could you give me a hint?

Answer 1

start with zero, the question is then the same as $$2^0+2^1+2^2+...+2^n=2^{n+1}-1$$ The binary for the LHS is $$ 11111\dots 1 $$ with $n$ ones. add $1$ to get $$ 10000\dots 0, $$ with $n$ zeros in binary. This is the binary for $2^{n+1}$ and are done.

Answer 2

It is straightforward

$$2+2^2+\cdots+2^n+2^{n+1}=(2+2^2+\cdots+2^n)+2^{n+1}$$

Use the induction assumption

$$\begin{align} 2+2^2+\cdots+2^n+2^{n+1}&=2^{n+1}-2+2^{n+1}\\&=2\cdot 2^{n+1}-2\\&=2^{n+2}-2\end{align}$$

Answer 3

Define $S=2^1+\cdots+2^n$ and realize that $2S=2^2+\cdots+2^{n+1}=S+2^{n+1}-2$.