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

In summary: 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.
  • #1
Math325
1
0

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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  • #2
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.
 
  • #3
If you don't have to use induction, you can let A = the sum, and see what 2A looks like. 2A-A=A.
 
  • #4
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!
 

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

1. How does proof by induction work?

Proof by induction is a mathematical proof technique used to prove that a statement is true for all natural numbers. It involves proving that the statement holds true for the first natural number, and then showing that if it holds true for any arbitrary natural number, it also holds true for the next natural number. By repeating this process, the statement is proven to be true for all natural numbers.

2. What is the sum of 2^n - 1?

The sum of 2^n - 1 is a mathematical series that starts with 1 and increases by powers of 2, with each subsequent term being 2 times the previous term plus 1. For example, the first few terms are 1, 3, 7, 15, 31, and so on.

3. How is proof by induction used to prove the sum of 2^n - 1?

To prove the sum of 2^n - 1, we can use proof by induction to show that the statement holds true for n = 1, and then show that if it holds true for n = k, it also holds true for n = k+1. This will prove that the statement is true for all natural numbers, including n = 1.

4. What are the steps for using proof by induction to prove the sum of 2^n - 1?

The steps for using proof by induction to prove the sum of 2^n - 1 are as follows:

  1. Show that the statement holds true for n = 1.
  2. Assume that the statement holds true for n = k.
  3. Show that the statement also holds true for n = k+1.
  4. Conclude that the statement is true for all natural numbers by the principle of mathematical induction.

5. Can proof by induction be used to prove other mathematical statements?

Yes, proof by induction can be used to prove various mathematical statements, as long as they can be formulated in terms of natural numbers. It is a powerful and widely used technique in mathematics and is often used to prove theorems and solve problems in various fields such as algebra, calculus, and number theory.

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