Given a1 = 1/2(a0+A/a0),a2 = 1/2(a1+A/a1) and an+1 = 1/2(an+A/an) for n ≥ 2, where a > 0, A > 0.

Question

Given a₁ = \(\frac{1}2(a_0+\frac{A}{a_0}),a_2\) = \(\frac{1}2(a_1+\frac{A}{a_1})\) and a_n+1 = \(\frac{1}2(a_n+\frac{A}{a_n})\) for n ≥ 2, where a > 0, A > 0

Answer 1

Given, a₁ = \(\frac{1}2(a_0+\frac{A}{a_0}),a_2\) = \(\frac{1}2(a_1+\frac{A}{a_1})\) and a_n+1 = \(\frac{1}2(a_n+\frac{A}{a_n})\),a,A>0

Step1: For n = 1

As LHS = RHS. 

So, it is true for P(1) 

For n = k, let P(k) be true.

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

P(k+1):

As L.H.S = R.H.S 

Thus, P(k+1) is true. So, by the principle of mathematical induction 

P(n) is true for all n.