Induction proof to find formula

I ran into some problem when I am doing some review. I need to find the formula for the following by exploring the cases n = 1,2,3,4 and prove by induction

I have this sequence

$$a_n = 1/(1*2) + 1/(2*3) + 1/(3*4) + ...... + 1/n(n+1)$$

I know how to do it when they give you the formula, but how will you find the formula for this example?

help will be appreciated

Thanks

Since $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$, we have that $a_n=\sum_{k=1}^{n}\frac{1}{k(k+1)}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\dots+\frac{1}{n}-\frac{1}{n+1}=1-\frac{1}{n+1}$.

The general trick is the partial fraction decomposition. See the wikipedia: http://en.wikipedia.org/wiki/Partial_fraction_decomposition.

Showing a function is bijective and finding its inverse

Limit of integral with a Bessel function

Proving $\prod_{i=1}^n (2i-1)$ = $\frac{(2n)!}{2^nn!}$ for all natural numbers

A holomorphic function which has $|f(z)|>1$ for every $|z|=1$ and $|f(0)|<1$ has a fixed point inside the unit circle

The existence of bounded linear functional on a complex Hilbert space

Find $\{a, b, c\} \in \Bbb{Z}$ that minimise $\left\|v_0 + a\cdot v_1 + b\cdot v_2 + c\cdot v_3 \right\|$ where $\{v_0, v_1, v_2, v_3\} \in \Bbb{R}^3$

Finding the maximum and minimum values of $a^2\sin^2\theta + b^2\csc^2\theta$

sum of trace for representations of finite groups

Understanding proof that continuous image of connected is connected

Least Square Estimation of Linear Regression

How is it that $\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$ for all angles, even though the derivation holds only for $\cos A\cos B\neq 0$?