Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Principle of induction problem

  1. Oct 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Let x ∈ N. Show that there exists for each n ∈ N a natural number denoted by x^n (this is just a notation, but should tell you what we are doing)such that x^1 =x and x^σ(n) =x·x^n.

    2. Relevant equations
    σ(n) = n + 1

    3. The attempt at a solution

    So far my answer is:
    1. Let S = {n ∈ N | x^1 = x and x^σ(n) = x * x^n}.
    We know that x^1 = x for any x ∈ N and σ(1) = 1+1 = 2. So x^σ(n) = x^2 = x * x^1. Therefore 1 ∈ S. Now assume n ∈ S. We know that x^1 = x regardless of n. Since σ(n) = n + 1, it follows that σ(σ(n)) = σ(n) + 1 = n + 1 + 1. So x^σ(n) = x^(n+1+1) = x^(n+1) * x^1 = x * x^σ(n). Therefore σ(n) ∈ S. By the principle of induction we see that S = N.

    My question is: Do I need to go into more detail in showing x^1 = x for this to be a valid proof? and if so how can i do that?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?