How to prove a formula for the sum of powers of $2$ by induction?

How do I prove this by induction?

Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$

Here is my attempt.

Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true.

Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$

Our hypothesis is: $2^n = 2^{n+1} -1$

Here is where I'm getting off track. Lets look at the right side of the last equation: $2^{n+1} -1$ I can rewrite this as the following.

$2^1(2^n) - 1$ But, from our hypothesis $2^n = 2^{n+1} - 1$ Thus:

$2^1(2^{n+1} -1) -1$ This is where I get lost. Because when I distribute through I get this.

$2^{n+2} -2 -1$ This is wrong is it not? Am I not applying the rules of exponents correctly here? I have the solution so I know what I'm doing is wrong. Here is the correct proof.

Solution 1:

An easy way to do this is using binary. Here's an idea of what I mean:

• $2^0$ in binary is $1$
• $2^1$ in binary is $10$
• $2^2$ in binary is $100$

For a general rule:

$2^n$ in binary is $100\cdots0$ (n zeros)

Add those together and you get $2^0 + 2^1 + ... + 2^n$ in binary is $11...11$ ($n+1$ ones).

Now it's obvious that adding 1 to that gives you $$100\cdots00 \quad\text {($n+1$ zeros)}$$ Which as we all know is $2^{n+1}$.

Thus $2^{n+1} - 1$ is equal so the sum of powers of two up to $2^n$.

Solution 2:

Both

• Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$
• Our hypothesis is: $2^n = 2^{n+1} -1$

are wrong and should be

• Inductive Step to prove is: $ 2^0 + 2^1 + ... + 2^n + 2^{n+1} = 2^{n+2} - 1$
• Our hypothesis is: $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$

Add $2^{n+1}$ to both sides of the hypothesis and you have the step to prove since $2^{n+1}-1 +2^{n+1} = 2^{n+2} - 1$