Using principle of mathematical induction prove that √n < 1/√1+1/√2+1/√3

Question

Using principle of mathematical induction prove that

\(\sqrt{n}<\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+\)......+ \(\frac{1}{\sqrt{n}}\) for all natural numbers n ≥ 2.

Answer 1

Let P(n) =  \(\sqrt{n}<\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+\)......+ \(\frac{1}{\sqrt{n}}\) for all n ≥ 2.

Step1: For n=2, P(n):

Therefore, it is true for n=2. 

Step2: Let P(n) be true for n=k.

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

P(k+1):

so, LHS < RHS

So, it is true for n=k+1, thus by the principle of mathematical induction P(n) is true for all n ≥ 2