Question

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 $\frac{x-a}{y-a}=\frac{b}{y}$ ?

Answer 1

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 $\frac{b}{y}=\frac{b+(n-1)(a-b)}{a}...\left(2\right)$
Also y - a =$\frac{(b-a)a(n-1)}{b+(n-1)(a-b)}by\left(2\right)...\left(3\right)$
Hence $\frac{x-a}{y-a}=\frac{b+(n-1)(a-b)}{a}=\frac{b}{y}$