Question

If $x > 1, y > 1, z > 1$ are in GP, then $\frac{1}{1+In\, x}, \frac{1}{1+In\, y},\frac{1}{1+In\, z},$ are in

• A. AP
• B. HP
• C. GP
• D. none of these

Answer 1

Answer: (C)

GP

Answer 2

Let the common ratio of the GP be r. then
\hspace20mm y = xr and $z = xr^2$
$\Rightarrow In\, y = \, In\, x + \, In\, r\,$ and$\, In\, z = \, In \, x + 2\, In \, r$
Let \hspace10mm A =1 + In\, x, D = In\, r
Then, $\frac{1}{1+In\, x} =\frac{1}{A}, \frac{1}{1+In\, y}=\frac{1}{1+In\, x +\, In\, r}= \frac{1}{A+D}$
and $\, \, \, \frac{1}{1+In\, z} =\frac{1}{1+In\, x +2 \, In\, r}= \frac{1}{A+2D}$
Therefore, $\frac{1}{1+In\, x},\frac{1}{1+In\, y }, \frac{1}{1+In\, z}$ are in HP.