$\frac{n}{1 \cdot 2 \cdot 3} + \frac{n - 1}{2 \cdot 3 \cdot 4} + \frac{n - 2}{3 \cdot 4 \cdot 5} + \

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$\frac{n}{1 \cdot 2 \cdot 3} + \frac{n - 1}{2 \cdot 3 \cdot 4} + \frac{n - 2}{3 \cdot 4 \cdot 5} + \ldots\ldots\ldots$ upto $n$ terms is equal to

Accepted Answer

Sol. [Key Concept: $\frac{n - k + 1}{k(k + 1)(k + 2)}$ telescopes via partial fractions.]$$\sum_{r = 1}^{n}\mspace{2mu}\frac{n - (r - 1)}{r(r + 1)(r + 2)} = \frac{(n + 1)}{2}\sum_{r = 1}^{n}\mspace{2mu}\frac{1}{r(r + 1)} - \frac{1}{(r + 1)(r + 2)} - \sum_{r = 1}^{n}\mspace{2mu}\left( \frac{1}{r + 1} - \frac{1}{r + 2} ight)$$$$\begin{matrix} & \ = \frac{(n + 1)}{2}\left( \frac{1}{2} - \frac{1}{(n + 1)(n + 2)} ight) - \left( \frac{1}{2} - \frac{1}{n + 2} ight) \ & \ = \frac{1}{2(n + 2)} + \frac{n + 1}{4} - \frac{1}{2} \end{matrix}$$

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