The sequence $\left\{ a_{n} \right\}$ and $\left\{ b_{n} \right\}$ satisfy $a_{k}b_{k} = 1,k = 1,2,3

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The sequence $\left\{ a_{n} \right\}$ and $\left\{ b_{n} \right\}$ satisfy $a_{k}b_{k} = 1,k = 1,2,3,\ldots$. If the sum of first ' $n$ ' terms of $\left\{ a_{n} \right\}$ is equal to $\frac{n(n + 1)(n + 2)}{3}$. Then

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Sol. [Key Concept: Two coupled sequences with a partial-sum condition.]$$a_{n} = \frac{n(n + 1)(n + 2)}{3} - \frac{(n - 1)n(n + 1)}{3} = n(n + 1)$$$$\begin{matrix} & b_{n} = \frac{1}{n(n + 1)} \ & \ \sum_{r = 1}^{n}\mspace{2mu}\mspace{2mu}{	ext{ }b}_{r} = \sum_{r = 1}^{n}\mspace{2mu}\mspace{2mu}\left( \frac{1}{r} - \frac{1}{r + 1} ight) = 1 - \frac{1}{n + 1} = \frac{n}{n + 1} \end{matrix}$$✔ (a) $\sum_{r = 1}^{n}b_{r} = \frac{n}{n + 1}$✘ (b) $\lim = 1 
eq 2$✘ (c) not the partial sum✔ (d) $\lim_{n ightarrow \infty}\sum_{r = 1}^{n}b_{r} = 1$

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