If $\frac{1}{2 imes 3 imes 4}+\frac{1}{3 imes 4 imes 5}+\frac{1}{4 imes 5 imes 6}+\ldots

Question

If $\frac{1}{2 	imes 3 	imes 4}+\frac{1}{3 	imes 4 	imes 5}+\frac{1}{4 	imes 5 	imes 6}+\ldots+\frac{1}{100 	imes 101 	imes 102}$ $=\frac{k}{101}$, then $34 k$ is equal to

Professor PC Thomas Classes · Answer

(286) :\ $\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}+\ldots \ldots+\frac{1}{100 \cdot 101 \cdot 102}=\frac{k}{101}$\ $\Rightarrow \frac{4-2}{2 \cdot 3 \cdot 4}+\frac{5-3}{3 \cdot 4 \cdot 5}+\ldots . .+\frac{102-100}{100 \cdot 101 \cdot 102}=\frac{2 k}{101}$\ $\Rightarrow \frac{1}{2 \cdot 3}-\frac{1}{3 \cdot 4}+\frac{1}{3 \cdot 4}-\frac{1}{4 \cdot 5}+$\ $\ldots \ldots+\frac{1}{100 \cdot 101}-\frac{1}{101 \cdot 102}=\frac{2 k}{101}$\ $\Rightarrow \frac{1}{6}-\frac{1}{101 \cdot 102}=\frac{2 k}{101}$\ $\Rightarrow 2 k=\frac{101}{6}-\frac{1}{102}$\ $\Rightarrow 17 	imes 2 k=\frac{17 	imes 101}{6}-\frac{17}{102}$\ $=\frac{1717}{6}-\frac{1}{6}=\frac{1716}{6}$\ $\Rightarrow 34 k=286$\