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If x^(n) = a(0) + a(1) (1 + x) + a(2)(1 ...

If `x^(n) = a_(0) + a_(1) (1 + x) + a_(2)(1 + x)^(2) + .. .+ a_(n) (1 +x)^(n) = b_(0) + b_(1) (1 - x) + b_(2) (1 - x)^(2) + ... + b_(n)(1 - x)^(n)` then for n = 201, `(a_(101) , b_(101) )` is equal to:

A

`(-""^(201)C_(101),-""^(201)C_(101))`

B

`(""^(201)C_(101),-""^(201)C_(101))`

C

`(-""^(201)C_(101),""^(201)C_(101))`

D

`(""^(201)C_(101),""^(201)C_(101))`

Text Solution

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The correct Answer is:
B
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Knowledge Check

  • If x^(n)=a_(0)+a_(1)(1+x)+a_(2)(1+x)^(2)+….+a_(n)(1+x)^(n)=b_(0)+b_(1)(1-x)+b_(2)(1-x)^(2)+….+b_(n)(1-x)^(n) then for n=101, (a_(50),b_(50)) equals:

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    `(-""^(101)C_(50), ""^(101)C_(50))`
    B
    `(""^(101)C_(50), -""^(101)C_(50))`
    C
    `(-""^(101)C_(50), -""^(101)C_(50))`
    D
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  • If (1-x+x^(2))^(n) =a_(0) +a_(1) x +a_(2)x^(2) + …+ a_(2n)x^(2n) then a_(0) +a_(2) +a_(4) + …+ a_(2n) equals

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    `(3^(n)+1)/(2) `
    B
    `(3^(n)-1)/(2)`
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  • (1 + x + x^(2))^(n) = a_(0) +a_(1)x + a_(2) x^(2) +...+ a_(2n )x^(2n) , then a_(0) a_(2n) - a_(1) a_(2r+1) + a_(2) a_(2r+2) - a_(3) a_(2r+3) +...+a_(2n-2r)a_(2n)= .

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    0
    B
    1
    C
    `a_(n-r)`
    D
    `a_(n+r)`

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