Find largest number $N$

Question

Find the largest number $N$ such that: $$\sum_{n=5}^{N}{\frac{1}{n(n-2)}}<\frac{1}{4}$$

This was a question I came across while looking over local competition papers. It doesn't seem to follow any arithmetic or geometric sequence which I first thought of when first reading the problem.

So, I'm at a loss. How do you solve this?

Answer 1

Note that

$$ \frac1{n-2}-\frac1n=\frac2{n(n-2)}\;. $$

Thus, if you collect the even terms and the odd terms, most of the summands cancel (this is called “telescoping”), only one term involving $N$ remains in each partial sum, and you can get a simple inequality for $N$.

Answer 2

$$\frac{1}{n(n-2)}=\frac{1}{2 (n-2)}-\frac{1}{2 n}$$ which beautifully telescopes $$S_n=\sum_{n=5}^{N}{\frac{1}{n(n-2)}}=-\frac{1}{2 N}-\frac{1}{2 (N-1)}+\frac{7}{24}=\frac{7 N^2-31 N+12}{24 (N-1) N}$$ Now, you face a quadratic equation in $N$; solve it and use $\lfloor N\rfloor$.