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

AI Thread Summary
The discussion focuses on proving the formula 1 + 2 + 2^2 + ... + 2^(n-1) = 2^n - 1 using mathematical induction. Participants emphasize the need to establish a base case, typically n = 1, and then assume the formula holds for n = k before proving it for n = k + 1. Clarification is sought on what is meant by "similar bases," which seems to cause confusion. The importance of using the assumption from the induction step to complete the proof is highlighted. The thread ultimately reinforces the structured approach required for mathematical induction proofs.
Math325
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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!
 
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Math325 said:

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.
 
If you don't have to use induction, you can let A = the sum, and see what 2A looks like. 2A-A=A.
 
I'm not sure what you mean by "similar bases". What similar bases are you referring to?

Math325 said:

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!
 

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