Question

If a, 1, c are in A.P and a, 2, c are in G.P, then for a, b, c to be H.P the value of b = __

Answer 1

a, b, c are in H.P.

b is H.M. = $\frac{2ac}{a+c}$ -------------(1)

as a, 1, c are in A.P. $\Rightarrow$ $\frac{a+c}{2}$ = 1

a + c = 2 -------------(2)

as a, 2, c are in G.P. $\Rightarrow$ $2^2$ = ac

ac = 4 ---------------(3)

substituting (2) & (3) in (1),

b = $\Rightarrow$ $\frac{2\left(4\right)}{2}$ = 4