Question

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​
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Answer 1

✪ᴀɴsᴡᴇʀ★

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

★ᴇxᴘʟᴀɪɴᴀᴛɪᴏɴ★

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),

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

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

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