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.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!
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.Math325 said:Homework Statement
1 + 2 + 2^2 + 2^3 + ...+2^(n-1) = 2^n - 1
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!
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.
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.
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.
The steps for using proof by induction to prove the sum of 2^n - 1 are as follows:
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.