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Question

If $x comma blank y comma$ and $z$ are $p to the power of t h end exponent comma blank q to the power of t h end exponent$ and $r to the power of t h end exponent$ terms respectively of an A.P. and also of a G.P., then $x to the power of y minus z end exponent y to the power of z minus x end exponent z to the power of x minus y end exponent$ is equal to

$x y z$
0
1
None of these

The correct answer is: 1

Given that $x comma blank y comma$ and $z$ are $p to the power of t h end exponent comma blank q to the power of t h end exponent$ and $r to the power of t h end exponent$ terms of an A.P.
$therefore blank x equals A plus open parentheses p minus 1 close parentheses D$
$y equals A plus open parentheses q minus 1 close parentheses D$
$z equals A plus open parentheses r minus 1 close parentheses D$
$rightwards double arrow x minus y equals open parentheses p minus q close parentheses D$
$y minus z equals open parentheses q minus r close parentheses D$
$z minus x equals open parentheses r minus p close parentheses D$
Where $A$ is the first term and $D$ is the common difference. Also $x comma blank y comma blank z$ are the $p to the power of t h end exponent comma blank q to the power of t h end exponent$ and $r to the power of t h end exponent$ terms of a G.P.
$therefore x equals a R to the power of p minus 1 end exponent comma blank y equals a R to the power of q minus 1 end exponent comma blank z equals a R to the power of r minus 1 end exponent$
$therefore x to the power of x minus z end exponent blank y to the power of z minus x end exponent blank z to the power of x minus y end exponent equals open parentheses a R to the power of p minus 1 end exponent close parentheses to the power of y minus z end exponent blank open parentheses a R to the power of q minus 1 end exponent close parentheses to the power of z minus x end exponent open parentheses a R to the power of r minus 1 end exponent close parentheses to the power of x minus y end exponent$
$equals a to the power of y minus z plus z minus x plus x minus y end exponent R to the power of open parentheses p minus 1 close parentheses open parentheses y minus z close parentheses plus open parentheses q minus 1 close parentheses open parentheses z minus x close parentheses plus open parentheses r minus 1 close parentheses open parentheses x minus y close parentheses end exponent$
$equals A to the power of 0 end exponent R to the power of open parentheses p minus 1 close parentheses open parentheses q minus r close parentheses D plus open parentheses q minus 1 close parentheses open parentheses r minus p close parentheses D plus open parentheses r minus 1 close parentheses left parenthesis p minus q right parenthesis D end exponent$
$equals A to the power of 0 end exponent R to the power of 0 end exponent equals 1$

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