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Question

The line $x plus y equals 1$ meets $x$ -axis at $A$ and $y$ -axis at $B comma blank P$ is the mid-point of $A B$ ;
$P subscript 1 end subscript$ is the foot of the perpendicular from $P$ to $O A$ :
$M subscript 1 end subscript$ is that of $P subscript 1 end subscript$ from $O P$ ; $P subscript 2 end subscript$ is that of $M subscript 1 end subscript$ from $O A$ ; $M subscript 2 end subscript$ is that of $P subscript 2 end subscript$ from $O P$ ; $P subscript 3 end subscript$ is that of $M subscript 2 end subscript$ from $O A$ ; and so on.
If $P subscript n end subscript$ denotes the $n to the power of t h end exponent$ foot of the perpendicular on $O A$ : then $O P subscript n end subscript$ is

${(\frac{1}{2})}^{n - 1}$
${(\frac{1}{2})}^{n}$
${(\frac{1}{2})}^{n + 1}$
None of these

The correct answer is: $open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of n end exponent$

We have,
$open parentheses O M subscript n minus 1 end subscript close parentheses to the power of 2 end exponent equals open parentheses O P subscript n end subscript close parentheses to the power of 2 end exponent plus open parentheses P subscript n end subscript M subscript n minus 1 end subscript close parentheses to the power of 2 end exponent$
$equals 2 open parentheses O P subscript n end subscript close parentheses to the power of 2 end exponent$
$equals 2 alpha subscript n end subscript superscript 2 end superscript$ (say)
Also,
$open parentheses O P subscript n minus 1 end subscript close parentheses to the power of 2 end exponent equals open parentheses O M subscript n minus 1 end subscript close parentheses to the power of 2 end exponent plus open parentheses P subscript n minus 1 end subscript M subscript n minus 1 end subscript close parentheses to the power of 2 end exponent$
$rightwards double arrow alpha subscript n minus 1 end subscript superscript 2 end superscript equals 2 alpha subscript n end subscript superscript 2 end superscript plus fraction numerator 1 over denominator 2 end fraction alpha subscript n minus 1 end subscript superscript 2 end superscript$
$rightwards double arrow alpha subscript n end subscript equals fraction numerator 1 over denominator 2 end fraction alpha subscript n minus 1 end subscript$
$rightwards double arrow O P subscript n end subscript equals alpha subscript n end subscript equals fraction numerator 1 over denominator 2 end fraction alpha subscript n minus 1 end subscript equals fraction numerator 1 over denominator 2 to the power of 2 end exponent end fraction alpha subscript n minus 1 end subscript equals horizontal ellipsis equals fraction numerator 1 over denominator 2 to the power of n end exponent end fraction equals open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of n end exponent$

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