How to find numbers (by induction) that make a sum big enough

Question

I have the sum $$A_n:=\sum_{k=n+1}^{2n}\frac1k $$

Now I have to find all numbers $n\in \mathbb{N},$ for which $A_n >\frac35$

How do I start the proof? I assume I have to use induction, but I have only done basic induction proofs to prove equalities.

Answer 1

By recalling that $H_n$ is defined as $\sum_{k=1}^{n}\frac{1}{k}$, we have $$ A_n = H_{2n}-H_n \tag{1}$$ hence: $$ A_{n+1}-A_n = \frac{1}{2n+2}+\frac{1}{2n+1}-\frac{1}{n+1} = \frac{1}{(2n+1)(2n+2)}\tag{2} $$ implying that the sequence $\{A_n\}_{n\geq 1}$ is increasing. By direct inspection of $A_1,A_2,A_3$, we have that $ A_n> \frac{3}{5}$ as soon as $n\geq 3$.