Mathematics - Sequence & Series Question with Solution | TestHub

Given,

$\frac{1}{2.3.4}+\frac{1}{3.4.5}+\dots .+\frac{1}{100.101.102}=\frac{k}{101}$

$\Rightarrow \frac{2}{2.3.4}+\frac{2}{3.4.5}+\dots .+\frac{2}{100.101.102}=\frac{2k}{101}$

$\Rightarrow \frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+\dots ..+\frac{102-100}{100.101.102}=\frac{2k}{101}$

$\Rightarrow \frac{1}{2\times 3}-\frac{1}{3\times 4}+\frac{1}{3\times 4}-\frac{1}{4\times 5}+\dots .+\frac{1}{100\times 101}-\frac{1}{101\times 102}=\frac{2k}{101}$

$\Rightarrow \frac{1}{2\times 3}-\frac{1}{101\times 102}=\frac{2k}{101}$

$\Rightarrow \frac{101}{2\times 3}-\frac{101}{101\times 102}=2k$

$\Rightarrow 2k=\frac{101}{6}-\frac{1}{102}$

$\Rightarrow 34k=286$