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Find-the-coefficient-of-x-50-in-the-expression-1-x-1000-2x-1-x-999-3x-2-1-x-998-1001x-1000-




Question Number 139332 by bemath last updated on 26/Apr/21
Find the coefficient of x^(50)  in  the expression (1+x)^(1000)  +2x(1+x)^(999) +  3x^2 (1+x)^(998) +...+1001x^(1000)
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{50}} \:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{1000}} \:+\mathrm{2x}\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{999}} + \\ $$$$\mathrm{3x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{998}} +…+\mathrm{1001x}^{\mathrm{1000}} \\ $$
Answered by mr W last updated on 26/Apr/21
=Σ_(n=1) ^(1001) nx^(n−1) (1+x)^(1001−n)   =Σ_(n=1) ^(1001) Σ_(k=0) ^(1001−n) n (((1001−n)),(k) )x^(k+n−1)   x^(k+n−1) =x^(50)  ⇒k+n−1=50 ⇒k=51−n≥0 ⇒n≤51  coef. of x^(50) :  Σ_(n=1) ^(51) n (((1001−n)),((51−n)) )
$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{1001}} {\sum}}{nx}^{{n}−\mathrm{1}} \left(\mathrm{1}+{x}\right)^{\mathrm{1001}−{n}} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{1001}} {\sum}}\underset{{k}=\mathrm{0}} {\overset{\mathrm{1001}−{n}} {\sum}}{n}\begin{pmatrix}{\mathrm{1001}−{n}}\\{{k}}\end{pmatrix}{x}^{{k}+{n}−\mathrm{1}} \\ $$$${x}^{{k}+{n}−\mathrm{1}} ={x}^{\mathrm{50}} \:\Rightarrow{k}+{n}−\mathrm{1}=\mathrm{50}\:\Rightarrow{k}=\mathrm{51}−{n}\geqslant\mathrm{0}\:\Rightarrow{n}\leqslant\mathrm{51} \\ $$$${coef}.\:{of}\:{x}^{\mathrm{50}} : \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{51}} {\sum}}{n}\begin{pmatrix}{\mathrm{1001}−{n}}\\{\mathrm{51}−{n}}\end{pmatrix} \\ $$
Commented by mr W last updated on 26/Apr/21
Commented by bemath last updated on 28/Apr/21
i got coefficient of x^(50)  in (1+x)^(1002)   = C _(50)^(1002)  = ((1002!)/(50! 952!))
$$\mathrm{i}\:\mathrm{got}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{50}} \:\mathrm{in}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{1002}} \\ $$$$=\:\mathrm{C}\:_{\mathrm{50}} ^{\mathrm{1002}} \:=\:\frac{\mathrm{1002}!}{\mathrm{50}!\:\mathrm{952}!} \\ $$
Commented by mr W last updated on 28/Apr/21
how did you get?
$${how}\:{did}\:{you}\:{get}? \\ $$

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