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Maths-

General

Easy

Question

If in the expansion of $left parenthesis 1 plus x right parenthesis to the power of n end exponent$ , the coefficient of $r$ th and $left parenthesis r plus 2 right parenthesis$ th term be equal, then $r$ is equal to

$2 n$
$\frac{2 n + 1}{2}$
$\frac{n}{2}$
$\frac{2 n - 1}{2}$

The correct answer is: $fraction numerator n over denominator 2 end fraction$

Given that, $blank to the power of n end exponent C subscript r minus 1 end subscript equals blank to the power of n end exponent C subscript r plus 1 end subscript$
$rightwards double arrow fraction numerator n factorial over denominator open parentheses n minus r plus 1 close parentheses open parentheses n minus r close parentheses open parentheses n minus r minus 1 close parentheses factorial open parentheses r minus 1 close parentheses factorial end fraction$
$equals fraction numerator n factorial over denominator open parentheses n minus r minus 1 close parentheses factorial open parentheses r plus 1 close parentheses open parentheses r close parentheses open parentheses r minus 1 close parentheses factorial end fraction$
$rightwards double arrow r to the power of 2 end exponent plus r equals n to the power of 2 end exponent minus n r plus n minus n r plus r to the power of 2 end exponent minus r$
$rightwards double arrow n to the power of 2 end exponent minus 2 n r minus 2 r plus n equals 0$
$rightwards double arrow open parentheses n minus 2 r close parentheses open parentheses n plus 1 close parentheses equals 0 blank rightwards double arrow blank r equals fraction numerator n over denominator 2 end fraction$

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