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Question Number 73052 Answers: 1 Comments: 0

let P_n =X^n +X^(n−1) +....+X^2 +X−1 ∈R[X] 1)prove that P_n have one root x_n inside ]0,+∞[ 2)study the sequence x_n

$${let}\:{P}_{{n}} ={X}^{{n}} \:+{X}^{{n}−\mathrm{1}} \:+....+{X}^{\mathrm{2}} \:+{X}−\mathrm{1}\:\in{R}\left[{X}\right] \\ $$$$\left.\mathrm{1}\left.\right){prove}\:{that}\:{P}_{{n}} {have}\:{one}\:{root}\:{x}_{{n}} \:{inside}\:\right]\mathrm{0},+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{sequence}\:{x}_{{n}} \\ $$

Question Number 73051 Answers: 0 Comments: 1

factorize inside R[X] 1)X^5 −1 2)X^6 +1

$${factorize}\:{inside}\:{R}\left[{X}\right] \\ $$$$\left.\mathrm{1}\right){X}^{\mathrm{5}} −\mathrm{1}\:\: \\ $$$$\left.\mathrm{2}\right){X}^{\mathrm{6}} \:+\mathrm{1} \\ $$

Question Number 73049 Answers: 0 Comments: 0

simplifyA_n = Σ_(k=0) ^n ((k/n)−α)^2 C_n ^k X^k (1−X)^(n−k)

$${simplifyA}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\frac{{k}}{{n}}−\alpha\right)^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \:{X}^{{k}} \left(\mathrm{1}−{X}\right)^{{n}−{k}} \\ $$

Question Number 73048 Answers: 0 Comments: 0

solve inside Z^2 x^2 +3xy−2y^2 =122

$${solve}\:{inside}\:{Z}^{\mathrm{2}} \:\:{x}^{\mathrm{2}} \:+\mathrm{3}{xy}−\mathrm{2}{y}^{\mathrm{2}} \:=\mathrm{122} \\ $$

Question Number 73047 Answers: 1 Comments: 0

solve inside Z^3 x^2 +y^2 +z^(2 ) =2xyz

$${solve}\:{inside}\:{Z}^{\mathrm{3}} \:\:\:\:\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}\:} =\mathrm{2}{xyz} \\ $$

Question Number 73046 Answers: 1 Comments: 0

prove that ∀n ∈N Σ_(k=0) ^n k C_(2n) ^(n+k) =nC_(2n−1) ^n

$${prove}\:{that}\:\forall{n}\:\in{N}\:\:\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{\mathrm{2}{n}} ^{{n}+{k}} \:={nC}_{\mathrm{2}{n}−\mathrm{1}} ^{{n}} \\ $$

Question Number 73045 Answers: 0 Comments: 0

calculate Σ_(k=0) ^n (C_n ^k /C_(2n−1) ^k )

$${calculate}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{{C}_{{n}} ^{{k}} }{{C}_{\mathrm{2}{n}−\mathrm{1}} ^{{k}} } \\ $$

Question Number 73044 Answers: 0 Comments: 0

prove that Σ_(k=1) ^n H_k =(n+1)H_n −n and Σ_(k=1) ^n H_k ^2 =(n+1)H_n ^2 −(2n+1)H_n +2n

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} =\left({n}+\mathrm{1}\right){H}_{{n}} −{n} \\ $$$${and}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:−\left(\mathrm{2}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n} \\ $$

Question Number 73043 Answers: 1 Comments: 2

prove that H_n =Σ_(k=1) ^n (((−1)^(k+1) )/k)×C_n ^k H_n =Σ_(k=1) ^n (1/k)

$${prove}\:{that}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}}×{C}_{{n}} ^{{k}} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 73042 Answers: 1 Comments: 0

prove that for (n,p)∈N^★^2 Σ_(k=0) ^(p ) k C_n ^(p−k) C_n ^k =n C_(2n−1) ^(p−1) conclude the value of Σ_(k=0) ^n k (C_n ^k )^2

$${prove}\:{that}\:{for}\:\left({n},{p}\right)\in{N}^{\bigstar^{\mathrm{2}} } \:\:\:\sum_{{k}=\mathrm{0}} ^{{p}\:} \:{k}\:{C}_{{n}} ^{{p}−{k}} \:{C}_{{n}} ^{{k}} \:={n}\:{C}_{\mathrm{2}{n}−\mathrm{1}} ^{{p}−\mathrm{1}} \\ $$$${conclude}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} \\ $$

Question Number 73041 Answers: 1 Comments: 1

prove that ∀n∈ N Σ_(k=0) ^(2n) (−1)^k (C_(2n) ^k )^2 =(−1)^n C_(2n) ^n

$${prove}\:{that}\:\forall{n}\in\:{N}\:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\left(−\mathrm{1}\right)^{{k}} \:\left({C}_{\mathrm{2}{n}} ^{{k}} \right)^{\mathrm{2}} \:=\left(−\mathrm{1}\right)^{{n}} \:{C}_{\mathrm{2}{n}} ^{{n}} \\ $$

Question Number 73040 Answers: 1 Comments: 0

prove that ∀(n,p)∈N^★ ×N 1)Σ_(k=0) ^p (−1)^k C_n ^k =(−1)^p C_(n−1) ^p 2)∀(p,q)∈N^2 Σ_(k=0) ^p C_(p+q) ^k C_(p+q−k) ^(p−k) =2^p C_(p+q) ^p

$${prove}\:{that}\:\:\forall\left({n},{p}\right)\in{N}^{\bigstar} ×{N} \\ $$$$\left.\mathrm{1}\right)\sum_{{k}=\mathrm{0}} ^{{p}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$$$\left.\mathrm{2}\right)\forall\left({p},{q}\right)\in{N}^{\mathrm{2}} \:\:\:\:\sum_{{k}=\mathrm{0}} ^{{p}} \:{C}_{{p}+{q}} ^{{k}} \:{C}_{{p}+{q}−{k}} ^{{p}−{k}} \:\:=\mathrm{2}^{{p}} \:{C}_{{p}+{q}} ^{{p}} \\ $$

Question Number 73039 Answers: 1 Comments: 0

let U_n =(n/2) if n even and U_n =((n−1)/2) if n odd let f(n)=Σ_(k=0) ^n U_k prove that ∀(x,y)∈N^2 f(x+y)−f(x−y)=xy

$${let}\:{U}_{{n}} =\frac{{n}}{\mathrm{2}}\:{if}\:{n}\:{even}\:{and}\:{U}_{{n}} =\frac{{n}−\mathrm{1}}{\mathrm{2}}\:{if}\:{n}\:{odd}\:{let}\:{f}\left({n}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {U}_{{k}} \\ $$$${prove}\:{that}\:\forall\left({x},{y}\right)\in{N}^{\mathrm{2}} \:\:\:\:{f}\left({x}+{y}\right)−{f}\left({x}−{y}\right)={xy} \\ $$

Question Number 73037 Answers: 1 Comments: 1

Question Number 73036 Answers: 1 Comments: 3

calculate 1) Σ_(k=1) ^n k^2 (n+1−k) 2)Σ_(1≤i≤j≤n) ij

$$\left.{calculate}\:\mathrm{1}\right)\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{2}} \left({n}+\mathrm{1}−{k}\right) \\ $$$$\left.\mathrm{2}\right)\sum_{\mathrm{1}\leqslant{i}\leqslant{j}\leqslant{n}} \:{ij} \\ $$

Question Number 73035 Answers: 0 Comments: 0

prove that ∀n∈N^★ 2!4!....(2n)!≥{(n+1)!}^n

$${prove}\:{that}\:\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\:\mathrm{2}!\mathrm{4}!....\left(\mathrm{2}{n}\right)!\geqslant\left\{\left({n}+\mathrm{1}\right)!\right\}^{{n}} \\ $$

Question Number 73034 Answers: 1 Comments: 1

calculate U_n =Σ_(k=1) ^n (k/((k+1)!))

$${calculate}\:{U}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}}{\left({k}+\mathrm{1}\right)!} \\ $$

Question Number 73033 Answers: 1 Comments: 0

solve inside N^2 x(x+1)=4y(y+1)

$${solve}\:{inside}\:{N}^{\mathrm{2}} \:\:\:\:{x}\left({x}+\mathrm{1}\right)=\mathrm{4}{y}\left({y}+\mathrm{1}\right) \\ $$

Question Number 73032 Answers: 1 Comments: 0

find x from n / ∃n∈N^n and 1+x+x^2 +x^3 +x^4 =n^2

$${find}\:{x}\:{from}\:{n}\:\:/\:\exists{n}\in{N}^{{n}} \:\:\:\:{and}\:\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{3}} \:+{x}^{\mathrm{4}} ={n}^{\mathrm{2}} \\ $$

Question Number 73031 Answers: 0 Comments: 0

solve inside N^2 3x^3 +xy +4y^3 =349

$${solve}\:{inside}\:{N}^{\mathrm{2}} \:\:\:\mathrm{3}{x}^{\mathrm{3}} \:+{xy}\:+\mathrm{4}{y}^{\mathrm{3}} \:=\mathrm{349} \\ $$

Question Number 73030 Answers: 2 Comments: 0

Question Number 73029 Answers: 1 Comments: 0

prove that ∀(n,p,q)∈N^3 Σ_(k=0) ^n C_p ^k C_q ^(n−k) =C_(p+q) ^n conclude that Σ_(k=0) ^n (C_n ^k )^2 =C_(2n) ^n

$${prove}\:{that}\:\:\forall\left({n},{p},{q}\right)\in{N}^{\mathrm{3}} \:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{p}} ^{{k}} \:{C}_{{q}} ^{{n}−{k}} \:\:\:={C}_{{p}+{q}} ^{{n}} \\ $$$${conclude}\:{that}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} \:={C}_{\mathrm{2}{n}} ^{{n}} \\ $$

Question Number 73028 Answers: 2 Comments: 0

calculate Σ_(1≤i≤n and 1≤j≤n) min(i,j)

$${calculate}\:\sum_{\mathrm{1}\leqslant{i}\leqslant{n}\:{and}\:\mathrm{1}\leqslant{j}\leqslant{n}} \:\:{min}\left({i},{j}\right) \\ $$

Question Number 73027 Answers: 1 Comments: 0

x and y are reals(or complex) let put x^((0)) =1 ,x^((1)) =x x^((2)) =x(x−1).....x^((n)) =x(x−1)(x−2)...(x−n+1)prove that (x+y)^((n)) =Σ_(k=0) ^n C_n ^k x^((n−k)) y^((k))

$${x}\:{and}\:{y}\:{are}\:{reals}\left({or}\:{complex}\right)\:{let}\:{put}\:{x}^{\left(\mathrm{0}\right)} =\mathrm{1}\:,{x}^{\left(\mathrm{1}\right)} ={x} \\ $$$${x}^{\left(\mathrm{2}\right)} ={x}\left({x}−\mathrm{1}\right).....{x}^{\left({n}\right)} ={x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)...\left({x}−{n}+\mathrm{1}\right){prove}\:{that} \\ $$$$\left({x}+{y}\right)^{\left({n}\right)} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:{x}^{\left({n}−{k}\right)} {y}^{\left({k}\right)} \\ $$

Question Number 73021 Answers: 0 Comments: 0

Question Number 73017 Answers: 0 Comments: 1

find lim_(x→+∞) x(√(x^2 + 1))

$${find}\: \\ $$$$\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:\:{x}\sqrt{{x}^{\mathrm{2}} \:+\:\mathrm{1}}\: \\ $$

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