Question

If $P\left(n\right):\sqrt{n}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}},n\in N,$ then P(n) is true for all n ≥ _____________.

Answer 1

$P\left(n\right):\sqrt{n}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}};n\in N$
for n = 1,
$P\left(1\right):1<\frac{1}{\sqrt{1}}=1$
which is a false statement.

for n = 2,
$P\left(2\right):\sqrt{2}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}$
which is true

for n = 3
$P\left(3\right):\sqrt{3}<1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}$
which is again true
Hence, $P\left(n\right):\sqrt{n}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+......\mathrm{is}\mathrm{true}\mathrm{for}n\ge 2$.