Question

If x,y,z are in H.P and $a^x$ = $b^y$ = $c^z$ , then a,b,c are in

• A. A.P
• B. G.P
• C. H.P
• D. None of these

Answer 1

The correct option is B

G.P

Let $a^x$ = $b^y$ = $c^z$ = k

$a=K^{\frac{1}{x}}$, $b=K^{\frac{1}{y}}$, $c=K^{\frac{1}{z}}$

Given x,y,z are in HP

$\Rightarrow$ $\frac{1}{x}$,$\frac{1}{y}$,$\frac{1}{z}$ are in A.P

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

$\Rightarrow$ $K^{\frac{2}{y}}$ = $K^{\frac{1}{x}+\frac{1}{z}}$

$K^{\frac{2}{y}}$ = $K^{\frac{1}{x}}$ . $K^{\frac{1}{z}}$

${\left[K^{\frac{1}{y}}\right]}^2$ = $K^{\frac{1}{x}}$ . $K^{\frac{1}{z}}$

$b^2$ = a.c

So, a,b,c are in G.P