Question

If $\frac{1}{a+b},\frac{1}{2b},\frac{1}{b+c}$ are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.

Answer 1

Here, $\frac{1}{a+b},\frac{1}{2b}\mathrm{and}\frac{1}{b+c}\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}.$

$$\begin{gathered} \therefore 2\times \frac{1}{2b}=\frac{1}{a+b}+\frac{1}{b+c} \\ \Rightarrow \frac{1}{b}=\frac{b+c+a+b}{\left(a+b\right)\left(b+c\right)} \\ \Rightarrow \left(a+b\right)\left(b+c\right)=b\left(2b+a+c\right) \\ \Rightarrow ab+ac+b^2+bc=2b^2+ab+bc \\ \Rightarrow 2b^2-b^2=ac \\ \Rightarrow b^2=ac \\ \mathrm{Thus},\mathrm{a},\mathrm{b}\mathrm{and}\mathrm{c}\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \end{gathered}$$