Question

If $\frac{(x+y)}{2},y,\frac{(y+z)}{2}$ are in HP, then $x,y,z$ are in

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

Answer 1

The correct option is B

GP

Explanation for the correct option.

We know that, if $a,b,c$ are in HP then $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$. So,

$\begin{array}{rcl}\frac{2}{y}& =& \frac{2}{x+y}+\frac{2}{y+z}\\ \Rightarrow \frac{1}{y}& =& \frac{1}{x+y}+\frac{1}{y+z}\\ \Rightarrow \left(x+y\right)\left(y+z\right)& =& y\left[y+z+x+y\right]\\ & \Rightarrow & xy+xz+y^2+yz=2y^2+yz+xy\\ \Rightarrow y^2& =& xz\end{array}$

Therefore, $x,y,z$ are in G.P.

Hence, option B is correct.