maths-binomial theorem

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coefficient of x^39 in the expansion of [(1/x^2) + x^4)]^18

  • maths-binomial theorem -

    Use the general term

    term(n+1) = C(18,n) (1/x^2)^(18-n) (x^4)^n

    = C(18,n) (x^(-36 +2n) ) (x^4n)
    = c(18,n) x^(2n - 36)

    ahhh, trick question,
    to have a term with x^39 , 2n-36 = 39
    there is no whole number solution,
    so there is no term with x^39

    e.g. --- take the first few terms
    [(1/x^2) + x^4)]^18
    = (1/x^2)^18 + 18(1/x^2)^17 (x^4) + 153 (1/x^2)^16 (x^4)^2 + ..
    = x^-36 + 18x^-30 + 153x^-24 + ..
    the exponent go
    -36 -30 -24 .. missing the 39

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