Question
If the $(m + 1)^{\text{th}\text{~}},(n + 1)^{\text{th}\text{~}}$ and $(r + 1)^{\text{th}\text{~}}$ terms of an A.P. are in G.P. and $m,n,r$ are in H.P., then the ratio of the common difference to the first term in the A.P. is equal to
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Solution
Sol. [Key Concept: Terms of an A.P. that are in G.P. (and in H.P.) fix the ratio of common difference to first term.]
$$\begin{matrix} & & \ \ \ \ \ \ \ \ & \ (a + nd)^{2} & \ = (a + md)(a + rd) \\ & \ \Rightarrow & \ \ \ \ \ \ \ \ & \left( n^{2} - mr \right)d^{2} & \ = ad(r + m - 2n) \\ & \ \Rightarrow & \ \ \ \ \ \ \ \ & \frac{d}{a} & \ = \frac{\left( \frac{2mr}{n} - 2n \right)}{n^{2} - mr} \\ & \ \Rightarrow & \ \ \ \ \ \ \ \ & \frac{d}{a} & \ = - \frac{2}{n} \end{matrix}$$
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