Question

A sequence x₁, x₂, x₃, …. is defined by letting x₁ = 2 and x_k = \(\frac{X_{k-1}}n\) for all natural numbers k, k ≥ 2. Show that X_n = \(\frac{2}{n}\) for all n ϵ N

Answer 1

Given: A sequence x₁, x₂, x₃, …. is defined by letting x₁ = 2 and x_k = \(\frac{X_{k-1}}n\) for all natural number k,k ≥ 2.

Now, we need to show P(k+1) is true whenever P(k) is true. 

P(k+1):

So, it is true for n=k+1. 

Thus, by the principle of mathematical induction P(n) is true.