Find the value of $$\dfrac{1}{4 \times 5} + \dfrac{1}{5 \times 6} + \dfrac{1}{6 \times 7} + ....... + \dfrac{1}{10 \times 11}$$:

The value of $$\dfrac{1}{4 \times 5} + \dfrac{1}{5 \times 6} + \dfrac{1}{6 \times 7} + ....... + \dfrac{1}{10 \times 11}$$:

The terms can be written as

$$\Sigma\ \dfrac{1}{\left(n+3\right)\left(n+4\right)}$$

=> Where sigma means the summation of all terms where n = 1 to n = 7

$$\Sigma\ \dfrac{1}{\left(n+3\right)\left(n+4\right)}$$

=> $$\Sigma\ \dfrac{\left(n+4\right)-\left(n+3\right)}{\left(n+3\right)\left(n+4\right)}$$

=> $$\Sigma\ \left[\dfrac{1}{\left(n+3\right)}-\dfrac{1}{\left(n+4\right)}\right]$$

=> $$\dfrac{1}{4}-\dfrac{1}{5}$$ + $$\dfrac{1}{5}-\dfrac{1}{6}$$ + .....$$\dfrac{1}{10}-\dfrac{1}{11}$$

=> All other terms will get cancelled, and only 1/4 and -1/11 will left.

Thus, $$\dfrac{1}{4}-\dfrac{1}{11}=\dfrac{7}{44}$$