The sum to 30 terms of the series $\frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\frac{1}{3.4 .5}+\ldots$ is

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Professor PC Thomas Classes · Answer

$\mathrm{t}_{\mathrm{n}}=\frac{1}{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)}=\frac{\frac{1}{2}}{\mathrm{n}}-\frac{1}{\mathrm{n}+1}+\frac{\frac{1}{2}}{\mathrm{n}+2}$ \ $ =\frac{1}{2}\left(\frac{1}{n}-\frac{1}{n+1}ight)-\frac{1}{2}\left(\frac{1}{n+1}-\frac{1}{n+2}ight) $ \  Putnn $=1,2,3 \ldots$ $\  \begin{aligned} & \mathrm{S}_{\mathrm{n}}=\frac{1}{2}\left(1-\frac{1}{\mathrm{n}+1}ight)-\frac{1}{2}\left(\frac{1}{2}-\frac{1}{\mathrm{n}+2}ight) \ & \mathrm{S}_{30}=\frac{1}{2}\left(1-\frac{1}{31}ight)-\frac{1}{2}\left(\frac{1}{2}-\frac{1}{32}ight) \end{aligned} $