First two terms of an A.P. as well as an H.P. are a and b. If x be any term of the A.P. and y the corresponding term of H.P., then will $\frac{x - a}{y - a} = \frac{b}{y}$ ?

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Solution

Yes. x = a + (n - 1)(b - a)
and $\frac{1}{y} = \frac{1}{a} (n - 1) [\frac{1}{b} - \frac{1}{a}]$
or x - a =(n - 1)(b - a) ....(1)
and $\frac{b}{y} = \frac{b + (n - 1) (a - b)}{a} . . . (2)$
Also y - a = $\frac{(b - a) a (n - 1)}{b + (n - 1) (a - b)} b y (2) . . . (3)$
Hence $\frac{x - a}{y - a} = \frac{b + (n - 1) (a - b)}{a} = \frac{b}{y}$

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