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Maths-

General

Easy

Question

The coefficient of $x to the power of negative n end exponent$ in $left parenthesis 1 plus x right parenthesis to the power of n end exponent open parentheses 1 plus fraction numerator 1 over denominator x end fraction close parentheses to the power of n end exponent$ is

0
1
$2^{n}$
${}^{2 n}C_{n}$

hint Hint:

The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
(a+b)ⁿ= ∑ⁿr=0ⁿCr a^n-rb^r,
where n is a positive integer and a, b are real numbers, and 0 < r ≤ n.

The correct answer is: 1

Given :
$left parenthesis 1 plus x right parenthesis to the power of n end exponent open parentheses 1 plus fraction numerator 1 over denominator x end fraction close parentheses to the power of n end exponent$

Further simplify
$left parenthesis 1 plus x right parenthesis to the power of n open parentheses 1 plus 1 over x close parentheses to the power of n space equals left parenthesis 1 plus x right parenthesis to the power of n open parentheses fraction numerator x space plus 1 over denominator x end fraction close parentheses to the power of n space equals space left parenthesis 1 plus x right parenthesis to the power of 2 n end exponent space. space x to the power of negative n end exponent left parenthesis 1 plus x right parenthesis to the power of 2 n end exponent space equals space left parenthesis x space plus space a right parenthesis to the power of n$

Binomial formula
$left parenthesis 1 plus x right parenthesis to the power of 2 n end exponent space equals space C presuperscript 2 n end presuperscript subscript 0 space left parenthesis 1 right parenthesis to the power of n plus space C presuperscript 2 n end presuperscript subscript 1 space left parenthesis 1 right parenthesis to the power of n minus 1 end exponent left parenthesis x right parenthesis to the power of 1 space plus C presuperscript 2 n end presuperscript subscript 2 left parenthesis 1 right parenthesis to the power of n minus 2 end exponent space left parenthesis x right parenthesis squared space plus space... space... x to the power of negative n end exponent left parenthesis 1 plus x right parenthesis to the power of 2 n end exponent space equals space x to the power of negative n end exponent open square brackets C presuperscript 2 n end presuperscript subscript 0 space plus space C presuperscript 2 n end presuperscript subscript 1 space left parenthesis x right parenthesis to the power of blank space plus C presuperscript 2 n end presuperscript subscript 2 space left parenthesis x right parenthesis squared space plus space... space... close square brackets M u l t i p l y i n g space x to the power of negative n space end exponent i n s i d e space t h e space b r a c k e t s comma x to the power of negative n end exponent left parenthesis 1 plus x right parenthesis to the power of 2 n end exponent space equals space x to the power of negative n end exponent space. C presuperscript 2 n end presuperscript subscript 0 space plus space C presuperscript 2 n end presuperscript subscript 1 space left parenthesis x right parenthesis to the power of 1 minus n end exponent space plus C presuperscript 2 n end presuperscript subscript 2 space left parenthesis x right parenthesis to the power of 2 minus n end exponent space plus space... space... T h e r e f o r e space comma space C o e f f i c i e n t space o f space x to the power of negative n end exponent space i s space C presuperscript 2 n end presuperscript subscript 0 W e space k n o w space C presuperscript n subscript 0 space equals space 1 T h e r e f o r e space C presuperscript 2 n end presuperscript subscript 0 space equals space 1$

Related Questions to study

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