Q: Prove $\displaystyle H_{1}+H_{2}+...+H_{n}=(n+1)H_{n}-n$.

A: First of all, this is only true for $\displaystyle n>1$. So, I let $\displaystyle n=2$ for my base case. The equation held, so I moved on.

Now, I assume $\displaystyle H_{1}+H_{2}+...+H_{k}=(k+1)H_{k}-k$ for some natrual number $\displaystyle k$.

Next, I have that

$\displaystyle H_{1}+H_{2}+...+H_{k}+H_{k+1}=((k+1)H_{k}-k)+H_{k+1}$ by our inductive hypothesis.

I expanded everything on the RHS to get

$\displaystyle (k+1)(1+\frac{1}{2}+...+\frac{1}{k})-k+(1+\frac{1}{2}+...+\frac{1}{k}+\frac{1}{k+1})$

$\displaystyle (k+1)+\frac{k+1}{2}+...+\frac{k+1}{k}-k+(1+\frac{1}{2}+...+\frac{1}{k}+\frac{1}{k+1})$

$\displaystyle (k+2)+\frac{k+2}{2}+...+\frac{k+2}{k}-k+\frac{1}{K+1}$

$\displaystyle (k+2)(1+\frac{1}{2}+...+\frac{1}{k})-k+\frac{1}{k+1}$

Now, I am not sure what to do. I think I may be taking the wrong approach to the problem. Any help would be great.

Thanks