Proof by Induction: Sum of 2^n - 1 Explained Step by Step

[Math325]

Homework Statement

1 + 2 + 2^2 + 2^3 + ...+2^(n-1) = 2^n - 1

Homework Equations

Sum formula?

3. The Attempt at a Solution
I know you are suppose to use the base case and prove its true for k +1, but I'm not sure what to do for similar bases. Can someone please solve it step by step? Thanks!

 

[Mark44]

Homework Statement

1 + 2 + 2^2 + 2^3 + ...+2^(n-1) = 2^n - 1

Homework Equations

Sum formula?

3. The Attempt at a Solution
I know you are suppose to use the base case and prove its true for k +1, but I'm not sure what to do for similar bases. Can someone please solve it step by step? Thanks!

No, we won't "solve" it for you step by step. The Physics Forums rules expressly forbid this. Anyway, you're not "solving" this problem, you need to prove that the statement above is true for all integers n, n ≥ 1.

For an induction proof, you need to do three things.
1. Show that the statement is true for n = 1 or whatever the base case is. Do you know what statement you need to prove when n = 1?
2. Assume that the statement is true for n = k. What does this statement look like?
3. Show that the statement is true for n = k + 1. You will need to use the assumption of step 2 to do this.

 

[RUber]

If you don't have to use induction, you can let A = the sum, and see what 2A looks like. 2A-A=A.

 

[HallsofIvy]

I'm not sure what you mean by "similar bases". What similar bases are you referring to?

Homework Statement

1 + 2 + 2^2 + 2^3 + ...+2^(n-1) = 2^n - 1

This says that 1= 2^1- 1= 2- 1, 1+ 2= 2^2- 1= 4- 1= 3 , 1+ 2+ 4= 2^3- 1= 8- 1= 7, etc.
The "base case" is, of course, when n= 1. Is "1= 2^1- 1" true?
Now, suppose 1+ 2+ 4+ 8+ ...+ 2^(n-1)= 2^n- 1. Write 1+ 2+ 4+ 8+ ...+ 2^(n-1)+ 2^n= (1+ 2+ 4+ 8+ ...+ 2^(n-1))+ 2^n

2. Homework Equations

Sum formula?

3. The Attempt at a Solution
I know you are suppose to use the base case and prove its true for k +1, but I'm not sure what to do for similar bases. Can someone please solve it step by step? Thanks!