Is there a way to show the following equality without induction I wanted to show the following equality without using induction:
$$
\sum_{k=2}^n \frac{1}{k(k-1)} = \frac{n-1}{n}
$$
Any hint on how to do it?

Hint: Partial fraction decomposition yields $$\frac{1}{k(k-1)} = \frac{1}{k-1} - \frac{1}{k}$$ which makes your series a special little beast called a telescoping series, try writing out the first few terms and see if you can spot a pattern that allows cancelling. 
In particular, you get $$\sum_{k=2}^n \frac{1}{k(k-1)} = \sum_{k=2}^n \left(\frac{1}{k-1} - \frac{1}{k}\right)$$ Which, when written out term by term looks something like $$= \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{n-2} - \frac{1}{n-1}\right) + \left(\frac{1}{n-1} - \frac{1}{n}\right)$$
Can you spot what cancels out?
