if three numbers are in HP then the numbers obtained by subtracting half of the middle term from each of ther are in:-

(a) AP
(b) GP
(c) HP
(d) none of these

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Solution

$We are given that three numbers are in HP & have to find the numbers obtained by subtracting half of the middle term from each of them . Le us take 3 terms as, a - d, a, a + d which are in A . P . \Rightarrow \frac{1}{a - d}, \frac{1}{a}, \frac{1}{a + d} are in H . P subtracting half of the middle term from each term, we get \Rightarrow \frac{1}{a - d} - \frac{1}{2 a}, \frac{1}{a} - \frac{1}{2 a}, \frac{1}{a + d} - \frac{1}{2 a} \Rightarrow \frac{2 a - a + d}{2 a (a - d)}, \frac{1}{2 a}, \frac{2 a - a - d}{2 a (a + d)} \Rightarrow \frac{a + d}{2 a (a - d)}, \frac{1}{2 a}, \frac{a - d}{2 a (a + d)} Now, \frac{t_{2}}{t_{1}} = \frac{1}{2 a} \times \frac{2 a (a - d)}{(a + d)} = \frac{a - d}{a + d} \frac{t_{3}}{t_{2}} = \frac{a - d}{2 a (a + d)} \times 2 a = \frac{a - d}{a + d} Since, \frac{t_{2}}{t_{1}} = \frac{t_{3}}{t_{2}}, so resulting sequence forms a GP with common ratio \frac{a - d}{a + d} .$

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