IF Ck is the coefficients of x power k in the expansion of (1+x)²⁰⁰⁵ and if a,dare real numbers then 2005 sigma k=0(a+kd).Ck is​ plz t

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IF Ck is the coefficients of x power k in the expansion of (1+x)²⁰⁰⁵ and if a,dare real numbers then 2005 sigma k=0(a+kd).Ck is​
plz tell fast if you got its urgent

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Jade 2 months 2021-11-28T11:29:41+00:00 1 Answer 0 views 0

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    2021-11-28T11:30:53+00:00

    ᴀɴsᴡᴇʀ

    \boxed{\sum\limits_{k=0}^{2005}(a+kd).C_{k}=(2a+ 2005d) {2}^{2004}}

    ᴇxᴘʟɪɴɪɴ

    Using Binomial Theorem

    (1+x)²⁰⁰⁵ = ²⁰⁰⁵C₀ + ²⁰⁰⁵C₁x + ²⁰⁰⁵C₂x² + ...²⁰⁰⁵C₂₀₀₅x²⁰⁰⁵

    ⇒(1+x)²⁰⁰⁵=\sum\limits_{k=0}^{2005} C_{k}{x} ^{k}....(1)

    Putting x = 1 in (1),we get

    \sum\limits_{k=0}^{2005} C_{k}=2²⁰⁰⁵....(2)

    Differenting (1) w.r.t x we get

    2005(1+x)²⁰⁰⁴ =\sum\limits_{k=0}^{2005} k.C_{k}{x} ^{k-1}

    Putting x = 1, we get

    \sum\limits_{k=0}^{2005} k.C_{k} =2005.2²⁰⁰⁴....(3)

    Now

    \sum\limits_{k=0}^{2005}(a+kd).C_{k}

    =a\sum\limits_{k=0}^{2005} C_{k} + d\sum\limits_{k=0}^{2005} k.C_{k}

    Using (2) and (3),

    =a.{2}^{2005} + d.2005.{2}^{2004}

    =2a.{2}^{2004} + d.2005.{2}^{2004}

    =(2a+ 2005d) {2}^{2004}

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