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If (1+x+x^2+x^3)^n=sum(r=0)^(3n)br x^r ...

If `(1+x+x^2+x^3)^n=sum_(r=0)^(3n)b_r x^r and sum_(r=0)^(3n)b_r=k ,"t h e n"sum_(r=0)^(3n)r b_r` is

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If (1+x+x^(2)+x^(3))^(n)=sum_(r=0)^(3n)b_(r)x^(r) and sum_(r=0)^(3n)b_(r)=k, the n sum_(r=0)^(3n)rb_(r)

If sum_(r=0)^(3n)a_(r)(x-4)^(r )=sum_(r=0)^(3n)A_(r)(x-5)^(r ) and a_( k)=1AA K ge 2n and sum_(r=0)^(3n)d_(r )(x-8)^(r )=sum_(r=0)^(3n)B_(r )(x-9)^(r ) and sum_(r =0)^(3n)d_(r )(x-12)^(r )=sum_(r=0)^(3n)D_(r )(x-13)^(r ) and d_(K)=1 AA K ge 2n . Then find the value of (A_(2n)+D_(2n))/(B_(2n)) .

Knowledge Check

  • Let (1 + x + x^(2))^(n) = sum_(r=0)^(2n) a_(r) x^(r) . If sum_(r=0)^(2n)(1)/(a_(r))= alpha , then sum_(r=0)^(2n) (r)/(a_(r)) =

    A
    n`alpha`
    B
    `(n - 1) alpha `
    C
    `(alpha)/(n)`
    D
    none of these

  • If (1+x)^(n)=sum_(r=0)^(n)*C_(r )x^(r ) and sum_(r=0)^(n) (-1)^(r ) (C_(r ))/((r+1)^(2)=k sum_(r=0)^(n)(1)/(r+1) , then k is equal to :

    A
    `(1)/(n) `
    B
    `(1)/(n+1)`
    C
    `(n)/(n+1)`
    D
    none

  • If (1+x)^(n)= sum_(r=0)^(n)C_(r )x^(r ) and sum_(r =0)^(n) (C_(r ))/(r+1)=k then the value of k is

    A
    `(2^(n+1)+1)/(n+1)`
    B
    `(2^(n+1)-1)/(n+1)`
    C
    `(2^(n)+1)/(n+1)`
    D
    `(2^(n)-1)/(n+1)`

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    If sum_(r=0)^(3n)a_(r)(x-4)^(r )=sum_(r=0)^(3n)A_(r)(x-5)^(r ) and a_( k)=1AA K ge 2n and sum_(r=0)^(3n)d_(r )(x-8)^(r )=sum_(r=0)^(3n)B_(r )(x-9)^(r ) and sum_(r =0)^(3n)d_(r )(x-12)^(r )=sum_(r=0)^(3n)D_(r )(x-13)^(r ) and d_(K)=1 AA K ge 2n . Then find the value of (A_(2n)+D_(2n))/(B_(2n)) .

    If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(n) and sum_(r=0)^(n)(C_(r))/(r+1)=k then the value of k is

    sum_(r=1)^(n)(3^(r)-r)=

    If ninN,ngt4 and (1-x^(3))^(n)=sum_(r=0)^(n)a_(r)x^(r)(1-x)^(3n-2r) , then sum_(r=0)^(n)a_(r)=lambda^(n) , find lambda.

    If (1+2x+x^(2))^(n)= sum_(r=0)^(2n) a_(r )x^(r ) , then a_(r )=

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