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Maths-

General

Easy

Question

For $x blank R blank element of comma blank x blank not equal to blank minus 1 comma$ if $left parenthesis 1 plus x right parenthesis to the power of 2016 end exponent plus x left parenthesis 1 plus x right parenthesis to the power of 2015 end exponent plus x to the power of 2 end exponent left parenthesis 1 plus x right parenthesis to the power of 2014 end exponent plus midline horizontal ellipsis plus x to the power of 2016 end exponent equals stretchy sum from i equals 0 to 2016 of a subscript i end subscript x to the power of i end exponent$ then $a subscript 17 end subscript blank$ is equal to

$\frac{2017!}{17! 2000!}$
$\frac{2016!}{17! 1999!}$
$\frac{2016!}{16!}$
$\frac{2017!}{2000!}$

The correct answer is: $fraction numerator 2017 factorial over denominator 17 factorial 2000 factorial end fraction$

We have
$table row cell left parenthesis 1 plus x right parenthesis to the power of 2016 end exponent fraction numerator open parentheses fraction numerator x over denominator 1 plus x end fraction close parentheses to the power of 2017 end exponent minus 1 over denominator fraction numerator 1 over denominator 1 plus x end fraction minus 1 end fraction equals stretchy sum from i equals 0 to 2016 of a subscript i end subscript x to the power of i end exponent end cell row cell left parenthesis 1 plus x right parenthesis to the power of 2017 end exponent fraction numerator left parenthesis 1 plus x right parenthesis to the power of 2017 end exponent minus x to the power of 2017 end exponent over denominator left parenthesis 1 plus x right parenthesis to the power of 2017 end exponent end fraction equals stretchy sum from i equals 0 to 2016 of a subscript i end subscript x to the power of i end exponent end cell end table$
$left parenthesis 1 plus x right parenthesis to the power of 2017 end exponent minus x to the power of 2017 end exponent equals stretchy sum from i equals 0 to 2016 of a subscript i end subscript x to the power of i end exponent$
Therefore,
$a subscript 17 end subscript equals text Coefficient of end text x to the power of 17 end exponent text in end text left parenthesis x plus 1 right parenthesis to the power of 2017 end exponent equals blank to the power of 2017 end exponent C subscript 17 end subscript equals fraction numerator 2017 factorial over denominator 17 factorial 2000 factorial end fraction$

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