What is the value of $$\frac{1}{3 \times 7} + \frac{1}{7 \times 11} + \frac{1}{11 \times 15} +.....+\frac{1}{899 \times 903}?$$

We have :
$$\frac{1}{3\times\ 7}+\frac{1}{7\times\ 11}+.....+\frac{1}{899\times\ 903}$$
Now we always use partial fraction in such sums 
Therefore we get 
$$\frac{1}{4}\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+........-\frac{1}{903}\right)$$
solving we get $$\frac{1}{4}\left(\frac{1}{3}-\frac{1}{903}\right)=\frac{1}{4}\left(\frac{300}{903}\right)=\frac{75}{903}=\frac{25}{301}$$