Question

If the $m^{th},n^{th}and{p}^{th}$ terms of A.P.& G.P be equal & are respectively $x,y,z$ then which of the following hold good

• A. $xyz=1$
• B. $x^y{y}^z{z}^x=x^z{y}^x{z}^y$
• C. $x^y{y}^z{z}^x=1$
• D. $x^z{y}^x{z}^y=1$

Answer 1

The correct option is B $x^y{y}^z{z}^x=x^z{y}^x{z}^y$

Let first term of $A.P$ be a, with common difference d
$\therefore x=a+(m-1)d=t_m$
$y=a+(n-1)d=t_n$
$z=a+(p-1)d=t_p$
Further say $A$ be the first term of $G.P$ with common ratio $R$:
$\therefore x=A{R}^{m-1}=t_m$
$y=A{R}^{n-1}=t_n$
$z=A{R}^{p-1}=t_p$
$\therefore x^{y-z}\cdot y^{z-x}\cdot z^{x-y}=A^{\circ }{R}^{\circ }=1$
$\Rightarrow x^y{y}^z{z}^x=x^z{y}^x{z}^y$
Ans: B