If $$A = \frac{1}{1 \times 2} + \frac{1}{1 \times 4} + \frac{1}{2 \times 3} + \frac{1}{4 \times 7} + \frac{1}{3 \times 4} + \frac{1}{7 \times 10}...$$ upto 20 terms, then what is the value of $$A?$$

A=$$\frac{1}{1 \times 2} + \frac{1}{1 \times 4} + \frac{1}{2 \times 3} + \frac{1}{4 \times 7} + \frac{1}{3 \times 4} + \frac{1}{7 \times 10}...$$ upto 20 terms

It has two series i.e $$\frac{1}{1 \times 2} + \frac{1}{2 \times 3}  + \frac{1}{3 \times 4} + ...\frac{1}{10 \times 11} $$and

 $$\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10}+....\frac{1}{28 \times 31}$$

=$$\frac{1}{1 } - \frac{1}{2}+ \frac{1}{2 }  + \frac{1}{3 } + ... \frac{1}{10}-\frac{1}{ 11} +\frac{1}{3} (\frac{1}{1} - \frac{1}{4} + \frac{1}{4}+....\frac{1}{28}-\frac{1}{31})$$

=(1-(1/11))+(1/3)(1-(1/31))

=10/11 +10/31
=420/31