Question : The sum of the first 20 term of the series $\frac{1}{5×6}+\frac{1}{6×7}+\frac{1}{7×8}+....$ is:
Option 1: 0.16
Option 2: 1.6
Option 3: 16
Option 4: 0.016
Correct Answer: 0.16
Solution :
Given: The sum of the first 20 terms of the series $\frac{1}{5×6}+\frac{1}{6×7}+\frac{1}{7×8}+....$
So, $a_{1} =5$ and $a_{20} =5+(20-1)×1=24$
20
th
term $=\frac{1}{24×25}$
$=\frac{1}{5×6}+\frac{1}{6×7}+\frac{1}{7×8}+.... + \frac{1}{24×25}$
$=\frac{1}{5}-\frac{1}{6} + \frac {1}{6} - \frac{1}{7}+\frac{1}{7}-\frac{1}{8}+..............+\frac{1}{24}-\frac{1}{25}$
$=\frac{1}{5}-\frac{1}{25}$
$=\frac{4}{25}$
$= 0.16$
Hence, the correct answer is 0.16.
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