The sum of the first 20 terms of the series $$\frac{1}{5×6}+\frac{1}{6×7}+\frac{1}{7×8}+...$$​ is:

Correct option is D

Given:

$$\frac{1}{5×6}+\frac{1}{6×7}+\frac{1}{7×8}+...$$

Concept Used:

Each term in the series can be rewritten as:

​$$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$$​​

Solution:

$$\frac{1}{5×6}+\frac{1}{6×7}+\frac{1}{7×8}+...$$

= $$\left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{8} \right) + \dots + \left( \frac{1}{24} - \frac{1}{25} \right)$$​​

That most terms cancel out:

= $$\frac{1}{5} - \frac{1}{25}$$

= $$\frac{5-1}{25}$$

= $$\frac{4}{25}$$

= 0.16

Thus, the correct answer is (d).