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

Answer: (B)

Let first term of $A.P$. be $a$, with common difference $d$
$\therefore x=a+\left(m-1\right)d=t_m$
$y=a+\left(n-1\right)d=t_n$
$z=a+\left(p-1\right)d=t_p$
Further say, $A$ be the first term of $GP$. with common ratio $R$.
$\therefore x=A{R}^{m-1}=t_m$
$y=A{R}^{n-1}=t_n$
$z=A{R}^{p-1}=t_p$
$\therefore x^{y-z}\cdot y^{z-x}\cdot z^{x-y}=A^0{R}^0=1$
$\therefore x^y{y}^z{z}^x=x^z{y}^x{z}^y$