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Question

If $S subscript n end subscript$ denotes the sum of first $n$ terms of an A.P. whose first term is $a$ and $fraction numerator S subscript n x end subscript over denominator S subscript x end subscript end fraction$ is independent of $x$ , then $S subscript p end subscript equals$

$p^{3}$
$p^{2} a$
$p a^{2}$
$a^{3}$

The correct answer is: $p to the power of 2 end exponent a$

$fraction numerator S subscript n x end subscript over denominator S subscript x end subscript end fraction equals fraction numerator fraction numerator n x over denominator 2 end fraction left square bracket 2 a plus left parenthesis n x minus 1 right parenthesis d right square bracket over denominator fraction numerator x over denominator 2 end fraction left square bracket 2 a plus left parenthesis x minus 1 right parenthesis d right square bracket end fraction$
$equals fraction numerator n left square bracket open parentheses 2 a minus d close parentheses plus n x d right square bracket over denominator open parentheses 2 a minus d close parentheses plus x d end fraction$
For $fraction numerator S subscript n x end subscript over denominator S subscript x end subscript end fraction$ to be independent of $x$ ,
$2 a minus d equals 0 blank rightwards double arrow 2 a equals d$
Now, $S subscript P end subscript equals fraction numerator p over denominator 2 end fraction open square brackets 2 a plus open parentheses p minus 1 close parentheses d close square brackets equals p to the power of 2 end exponent a$

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