Mathematics - Sequence & Series Question with Solution | TestHub

If $a,b,c$ are in Harmonic Progression (H.P.), then their reciprocals $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in Arithmetic Progression (A.P.).

Therefore, the common difference implies:

$$\frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}$$

From this, we can derive:

$$\left(\frac{1}{c}+\frac{1}{c}-\frac{1}{b}\right)\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a}\right)=\left(\frac{2}{c}-\frac{1}{b}\right)\left(\frac{1}{b}\right)=\frac{2}{bc}-\frac{1}{b^2}$$