Question

If a,b,c are in HP and half the middle term be subtracted from the three terms, show that the resulting series will be in GP

Answer 1

Since, a, b, c are in HP so

$$\begin{gathered} b=\frac{2ac}{a+c} \\ \Rightarrow b\left(a+c\right)=2ac \end{gathered}$$

If $\frac{b}{2}$ is subtracted from a, b, c then the terms becomes $a-\frac{b}{2},\frac{b}{2},c-\frac{b}{2}$.

If x, y, z are in GP then $y^2=xz$

Consider,

$$\begin{gathered} \left(a-\frac{b}{2}\right)\left(c-\frac{b}{2}\right) \\ =ac-\frac{bc}{2}-\frac{ab}{2}+\frac{b^2}{4} \\ =\frac{2ac-b\left(a+c\right)}{2}+\frac{b^2}{4} \\ =\frac{b\left(a+c\right)-b\left(a+c\right)}{2}+\frac{b^2}{4}[Since,b\left(a+c\right)=2ac] \\ =0+\frac{b^2}{4} \\ =\frac{b^2}{4}={\left(\frac{b}{2}\right)}^2 \end{gathered}$$

Thus, $a-\frac{b}{2},\frac{b}{2},c-\frac{b}{2}$ are in GP.