Question
If $\mathcal{l},m,n$ are three numbers in G.P., then the ratio of first term to the common difference of an A.P. whose $\mathcal{l}^{\text{th}\text{~}},m^{\text{th}\text{~}}$ and $n^{\text{th}\text{~}}$ terms are in H.P. is equal to
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Solution
Sol. [Key Concept: Three numbers in G.P., and an A.P. whose chosen terms are in H.P. ⇒ ratio of first term to common difference.]
$$\begin{matrix} & a + (m - 1)d = \frac{2(a + (\mathcal{l} - 1)d)(a + (n - 1)d)}{(a + (\mathcal{l} - 1)d + a + (n - 1)d)} \\ \Rightarrow \ & \ (2a + (\mathcal{l} + n - 2)d)(a + (m - 1)d) = 2(a + (\mathcal{l} - 1)d)(a + (n - 1)d) \\ \Rightarrow \ & ad\lbrack(2\text{ }m - 2 + \mathcal{l} + n - 2) - (2(n - 1) + 2(\mathcal{l} - 1))\rbrack \\ & \ = d^{2}(2(\mathcal{l} - 1)(n - 1) - (\mathcal{l} + n - 2)(m - 1)) \\ \Rightarrow \ & \frac{a}{\text{ }d} = \frac{2\mathcal{l}n - \mathcal{l} - n - \mathcal{l}m - mn + 2\text{ }m}{2\text{ }m - n - \mathcal{l}} \\ & \ = \frac{(2\text{ }m - \mathcal{l} - n)(m + 1)}{(2\text{ }m - n - \mathcal{l})} \\ & \ = (m + 1) \end{matrix}$$
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