In this equation, how does the left term simplify to the right term?

Question

I have obtained the following formula for some mathematical question:

$$\sum_{i=1}^{n-1} \frac{n(n-1)}{i(i+1)}.$$

I asked somehow who knew the question, and more importantly, the answer. He said the answer was $(n-1)^2$.

It would seem that, after testing for some values, my result is the same as his result. But I can't seem to simplify my formula to $(n-1)^2$, and neither can symbolab apparently. My question is thus, is the following equation correct, and if so, how does one simplify the left term to the right?

$$\sum_{i=1}^{n-1} \frac{n(n-1)}{i(i+1)} = (n-1)^2.$$

Answer 1

$\sum\limits_{i=1}^{n-1}\frac 1 {i(i+1)}=\sum\limits_{i=1}^{n-1} (\frac 1 i -\frac 1 {i+1})=1-\frac 1 n$ Now just multiply by $n(n-1)$.

Answer 2

$$ \sum_{i=1}^{n-1} \frac{n(n-1)}{i(i+1)}=n(n-1)\sum_{i=1}^{n-1} \left(\frac1 i-\frac1{i+1} \right)=n(n-1)\left(\frac11-\frac12+\frac12-\frac13+\cdots+\frac1{n-1} -\frac1n\right)$$ etc.