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  4. The sum of first 20 terms of (1/1 × 2) + (1/2 × 3) + (1/3 × 4)........ is

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Q. The sum of first $20$ terms of
$ \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4}........$ is

COMEDKCOMEDK 2008Sequences and Series

Solution:

$ \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4}...$
$T_{n} = \frac{1}{n\left(n+1\right)} =\frac{1}{n} - \frac{1}{n+1} $
$ \therefore \:\:\:T_{1 } = 1 - \frac{1}{2}$
$ T_{2} = \frac{1}{2}-\frac{1}{3} $
$T_{3} =\frac{1}{3}- \frac{1}{4}$
$ T_{20} = \frac{1}{20} - \frac{1}{21}$
On adding, we get
$S_{20} = 1- \frac{1}{21} = \frac{20}{21}$