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Question

If $a comma blank b comma$ and $c$ are in A.P., $p comma q$ and $r$ are in H.P. and $a p comma blank b q$ , and $c r$ are in G.P., then $fraction numerator p over denominator r end fraction plus fraction numerator r over denominator p end fraction$ is equal to

$\frac{a}{c} - \frac{c}{a}$
$\frac{a}{c} + \frac{c}{a}$
$\frac{b}{q} + \frac{q}{b}$
$\frac{b}{q} - \frac{q}{b}$

The correct answer is: $fraction numerator a over denominator c end fraction plus fraction numerator c over denominator a end fraction$

$fraction numerator p over denominator r end fraction plus fraction numerator r over denominator p end fraction equals fraction numerator p to the power of 2 end exponent plus r to the power of 2 end exponent over denominator p r end fraction equals fraction numerator open parentheses p plus r close parentheses to the power of 2 end exponent minus 2 p r over denominator p r end fraction$
$equals fraction numerator fraction numerator 4 p to the power of 2 end exponent r to the power of 2 end exponent over denominator q to the power of 2 end exponent end fraction minus 2 p r over denominator p r end fraction blank open square brackets table row cell because p comma blank q comma blank r blank a r e blank i n blank H. P. end cell row cell therefore q equals fraction numerator 2 p r over denominator p plus r end fraction end cell end table close square brackets$
$equals fraction numerator 4 p r over denominator q to the power of 2 end exponent end fraction minus 2 equals fraction numerator 4 b to the power of 2 end exponent over denominator a c end fraction minus 2 blank open square brackets because a p comma blank b q comma blank c r blank a r e blank i n blank A. P. blank rightwards double arrow b to the power of 2 end exponent q to the power of 2 end exponent equals a c p r close square brackets$
$equals fraction numerator open parentheses a plus c close parentheses to the power of 2 end exponent over denominator a c end fraction minus 2 blank open square brackets a comma blank b comma blank c comma blank a r e blank i n blank A. P. blank rightwards double arrow 2 b equals a plus c close square brackets$
$equals fraction numerator a over denominator c end fraction plus fraction numerator c over denominator a end fraction$

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