Question

If $1_{/a^x}=1{/}_{b^y}=1_{/c^z}$ and a, b, c are in G.P. Prove that x, y, z are in H.P.

Answer 1

We are given that

$\frac{1}{a^x}=\frac{1}{b^y}=\frac{1}{c^z}$ & $a,b,c$ are in $G.P$.

$\therefore a^x=b^y=c^z$------$\left(1\right)$ & $\frac{b}{a}=\frac{c}{b}\Rightarrow b^2=ac$-------$\left(2\right)$

now, taking logarithmic Function to be base $e$. We get,

$x{\log }_{e}a=y{\log }_{e}b=z{\log }_{e}c⟶\left(3\right)$

& ${\log }_e{b}^2={\log }_e(a.c)⟶\left(3\right)$

$2{\log }_{e}b={\log }_{e}a+{\log }_{e}c$

$2{\log }_{e}b=\frac{y}{x}{\log }_{e}b+\frac{y}{z}{\log }_{e}b$

( From eqn $\left(3\right)$ )

$\therefore 2=\frac{y}{x}+\frac{y}{z}$

$\therefore 2=y(\frac{1}{x}+\frac{1}{z})$

$\therefore \frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

So, $x,y$ & $z$ are in $H.P$.

Hence Prove that $x,y,z$ are in $HP$.