Please help me with the proof below!

Question

let ⟨x_n⟩_n=1...∞ be a sequence satisfying x_n+1=x_n-x_n-1 for each n.
Prove that ∀ n ∈ N, x_n+6=x_n by induction

 

 

Please help me. I'm reviewing for my final and something like this may appear on it.

Answer 1

Induction proofs require two steps.  First, you must show that the proof is true at lest once.  Then you show that if it is true for n = k, it is also true for n = k+1.

 

Let's start with the first step.  We need to show that x₇=x₁.

 

x₇= x₆ - x₅ = (x₅ - x₄) - x₅ = -x₄ = -(x₃ - x₂) = -[(x₂ - x₁₎ - x₂] = -(-x₁) = x₁

 

(Note: It's a little tedious, but keep working to apply the original sequence definition until you reach your goal.)

 

Now, move onto the second part.  Assume x_k+6=x_k is true for some k ∈ N.

 

We must prove the statement true for n = k+1.  (note: (k+1)+6 = k+7)

 

x_k+7 = x_k+6 - x_k+5 = x_k - x_k-1 = x_k+1

 

This completes the proof by induction.