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Question Number 219563 Answers: 2 Comments: 0

find all n ∈ N^∗ such that ∫_0 ^( 1) (sinx)^(2n−2) ∙ (cosx)^(2n) dx ≥ (1/4^(1011) )

$$\mathrm{find}\:\mathrm{all}\:\:\:\mathrm{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\left(\mathrm{sinx}\right)^{\mathrm{2n}−\mathrm{2}} \:\centerdot\:\left(\mathrm{cosx}\right)^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:\mathrm{dx}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{1011}} } \\ $$

Question Number 219562 Answers: 1 Comments: 0

prove that exists X ∈ M_(2,3) (R) Y ∈ M_(3,2) (R) such that X∙Y = ((1,1),(1,1) ) Y∙X = ((2,6,6),(3,9,9),((-3),(-9),(-9)) )

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{exists}\:\:\:\mathrm{X}\:\in\:\mathrm{M}_{\mathrm{2},\mathrm{3}} \:\left(\mathbb{R}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}\:\in\:\mathrm{M}_{\mathrm{3},\mathrm{2}} \:\left(\mathbb{R}\right) \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\:\mathrm{X}\centerdot\mathrm{Y}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}}\end{pmatrix}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}\centerdot\mathrm{X}\:=\:\begin{pmatrix}{\mathrm{2}}&{\mathrm{6}}&{\mathrm{6}}\\{\mathrm{3}}&{\mathrm{9}}&{\mathrm{9}}\\{-\mathrm{3}}&{-\mathrm{9}}&{-\mathrm{9}}\end{pmatrix}\: \\ $$

Question Number 219561 Answers: 0 Comments: 3

let be the sequence (x_n )n ≥ 1 defined by x_1 = 1 x_(n+2) = 3x_(n+1) − x_n ∀n ∈ N find L =lim_(n→∞) ((Σ_(k=0) ^0 (x_(2k+1) /(x_k + x_(k+1) ))))^(1/n) = ?

$$\mathrm{let}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sequence}\:\:\:\left(\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)\mathrm{n}\:\geqslant\:\mathrm{1} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\:\:\mathrm{x}_{\mathrm{1}} =\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}_{\boldsymbol{\mathrm{n}}+\mathrm{2}} \:=\:\mathrm{3x}_{\boldsymbol{\mathrm{n}}+\mathrm{1}} −\:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \\ $$$$\forall\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{find}\:\:\:\boldsymbol{\mathrm{L}}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\boldsymbol{\mathrm{n}}}]{\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\mathrm{0}} {\sum}}\:\:\frac{\mathrm{x}_{\mathrm{2}\boldsymbol{\mathrm{k}}+\mathrm{1}} }{\mathrm{x}_{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{x}_{\boldsymbol{\mathrm{k}}+\mathrm{1}} }}\:=\:? \\ $$

Question Number 219548 Answers: 0 Comments: 0

Question Number 219515 Answers: 2 Comments: 0

Find: 𝛀 = ∫_0 ^( 1) x^x^2 dx = ?

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\mathrm{dx}\:=\:?\: \\ $$

Question Number 219503 Answers: 1 Comments: 0

Question Number 219492 Answers: 3 Comments: 0

a + b = 1 a^2 + b^2 = 2 Find: a^(11) + b^(11) = ?

$$\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{1} \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:=\:\mathrm{2} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}^{\mathrm{11}} \:+\:\mathrm{b}^{\mathrm{11}} \:=\:? \\ $$

Question Number 219476 Answers: 1 Comments: 1

let a_1 = 1 ; (n+1)a_(n+1) +na_n = 2n−3 find nth term of a_(n )

$$\:\mathrm{let}\:\mathrm{a}_{\mathrm{1}} \:=\:\mathrm{1}\:;\:\left(\mathrm{n}+\mathrm{1}\right)\mathrm{a}_{\mathrm{n}+\mathrm{1}} +\mathrm{na}_{\mathrm{n}} \:=\:\mathrm{2n}−\mathrm{3}\: \\ $$$$\:\:\mathrm{find}\:\:\mathrm{nth}\:\mathrm{term}\:\mathrm{of}\:\mathrm{a}_{\mathrm{n}\:} \\ $$

Question Number 219459 Answers: 0 Comments: 0

Two cups m and n contains the same mass of water, m is at 25°c while n is at the temperature of 102°c. If both cups are placed in the same freezer of internal temperature - 45°c. Which of the content of m and n freezes first ? Hence, show that tₘ⁻ tₙ = - wc In(²¹/₁₀). where tₘ and tₙ are the time taken for m and n to freezes and w is the mass of water and c is specific heat capacity of water.

Question Number 219454 Answers: 1 Comments: 3

a, b, c are the roots of the equation x^3 −3x+1=0. find (a)^(1/3) +(b)^(1/3) +(c)^(1/3) =? & (1/( (a)^(1/3) ))+(1/( (b)^(1/3) ))+(1/( (c)^(1/3) ))=?

$${a},\:{b},\:{c}\:{are}\:{the}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${x}^{\mathrm{3}} −\mathrm{3}{x}+\mathrm{1}=\mathrm{0}. \\ $$$${find}\:\sqrt[{\mathrm{3}}]{{a}}+\sqrt[{\mathrm{3}}]{{b}}+\sqrt[{\mathrm{3}}]{{c}}=?\:\&\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{a}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{b}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{c}}}=? \\ $$

Question Number 219450 Answers: 1 Comments: 3

Question Number 219448 Answers: 0 Comments: 0

Question Number 219414 Answers: 0 Comments: 0

given the recursive {a_n } define by setting a_(1 ) ∈ (0,1) , a_(n+1) = a_n (1−a_n ) , n≥1 prove that (1) lim_(n→∞) na_n = 1 (2) b_n = n(1−na_n ) is a incresing sequence and diverge to ∞ (3) lim_(n→∞) ((n(1−na_n ))/(ln(n))) = 1

$$\:\:\:\mathrm{given}\:\mathrm{the}\:\mathrm{recursive}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}\:\mathrm{define}\:\mathrm{by}\:\mathrm{setting} \\ $$$$\:\:\mathrm{a}_{\mathrm{1}\:} \:\in\:\left(\mathrm{0},\mathrm{1}\right)\:\:\:,\:\:\:\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:=\:\mathrm{a}_{\mathrm{n}} \left(\mathrm{1}−\mathrm{a}_{\mathrm{n}} \right)\:\:\:,\:\mathrm{n}\geqslant\mathrm{1} \\ $$$$\:\:\mathrm{prove}\:\mathrm{that}\:\:\left(\mathrm{1}\right)\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{na}_{\mathrm{n}} =\:\mathrm{1} \\ $$$$\:\:\left(\mathrm{2}\right)\:\:\mathrm{b}_{\mathrm{n}} \:=\:\mathrm{n}\left(\mathrm{1}−\mathrm{na}_{\mathrm{n}} \right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{incresing}\:\mathrm{sequence} \\ $$$$\:\:\:\mathrm{and}\:\mathrm{diverge}\:\mathrm{to}\:\infty \\ $$$$\:\:\:\left(\mathrm{3}\right)\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{n}\left(\mathrm{1}−\mathrm{na}_{\mathrm{n}} \right)}{\mathrm{ln}\left(\mathrm{n}\right)}\:=\:\mathrm{1} \\ $$

Question Number 219404 Answers: 2 Comments: 2

given g(x)= ((x−2023)/(x−1)) find (gogogogogog)(2024)

$$\:\:\mathrm{given}\:\mathrm{g}\left(\mathrm{x}\right)=\:\frac{\mathrm{x}−\mathrm{2023}}{\mathrm{x}−\mathrm{1}} \\ $$$$\:\:\mathrm{find}\:\left(\mathrm{gogogogogog}\right)\left(\mathrm{2024}\right) \\ $$

Question Number 219343 Answers: 1 Comments: 0

∫∫∫_( S) e^(−x^2 −y^2 −z^2 ) dA=???

$$\int\int\int_{\:\mathrm{S}} \:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{z}^{\mathrm{2}} } \:\mathrm{dA}=??? \\ $$

Question Number 219352 Answers: 1 Comments: 0

Question Number 219351 Answers: 1 Comments: 0

Question Number 219315 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^4 ))=?

$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{4}} }=? \\ $$

Question Number 219304 Answers: 0 Comments: 0

Question Number 219281 Answers: 1 Comments: 0

Question Number 219283 Answers: 1 Comments: 0

Question Number 219267 Answers: 1 Comments: 0

Une fonction P est dite quasi polynomiale s′il existe (pour k∈N ) k+1 fonction periodique(c_i )_(i∈[∣0;k∣]) de Z dans R telles que P(n)=Σ_(k=1) ^n c_i (n)n^i (1) Montrez que l′ensemble des fonction quasi polynomiale forme un R−ev(real space vector). (2)Montrez que si P,Q:Z→R sont desfonction quasi polynomiale tel que P(n)=Q(n) ∀n∈N alors P=Q

$${Une}\:{fonction}\:{P}\:{est}\:{dite}\:{quasi}\:{polynomiale}\:{s}'{il}\:{existe}\:\left({pour}\:{k}\in\mathbb{N}\:\right)\:{k}+\mathrm{1}\:{fonction}\:{periodique}\left({c}_{{i}} \right)_{{i}\in\left[\mid\mathrm{0};{k}\mid\right]} {de}\:\mathbb{Z}\:{dans}\:\mathbb{R} \\ $$$$\:{telles}\:{que}\:{P}\left({n}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{c}_{{i}} \left({n}\right){n}^{{i}} \\ $$$$\left(\mathrm{1}\right)\:{Montrez}\:{que}\:{l}'{ensemble}\:{des}\:{fonction}\:{quasi}\:{polynomiale}\:{forme}\:{un}\:\mathbb{R}−{ev}\left({real}\:{space}\:{vector}\right). \\ $$$$\left(\mathrm{2}\right){Montrez}\:{que}\:{si}\:{P},{Q}:\mathbb{Z}\rightarrow\mathbb{R}\:{sont}\:{desfonction}\:{quasi}\:{polynomiale}\:{tel}\:{que}\:{P}\left({n}\right)={Q}\left({n}\right)\:\forall{n}\in\mathbb{N}\:{alors}\:{P}={Q} \\ $$

Question Number 219181 Answers: 3 Comments: 0

Question Number 219113 Answers: 3 Comments: 1

((a + 3b)/(a + b−1)) + ((a + 3b−1)/(a + b−3)) = 4 ⇒ a + b = ?

$$\frac{\mathrm{a}\:+\:\mathrm{3b}}{\mathrm{a}\:+\:\mathrm{b}−\mathrm{1}}\:+\:\frac{\mathrm{a}\:+\:\mathrm{3b}−\mathrm{1}}{\mathrm{a}\:+\:\mathrm{b}−\mathrm{3}}\:=\:\mathrm{4}\:\:\Rightarrow\:\:\mathrm{a}\:+\:\mathrm{b}\:=\:? \\ $$

Question Number 219112 Answers: 0 Comments: 0

Prove it: In triangle ABC, AB=c, BC=b, AC=a ab^2 c + abc^2 −a^2 bc ≥ tan (A/2) ((2S^3 )/((a+b)^2 −c^2 ))

$$\mathrm{Prove}\:\mathrm{it}: \\ $$$$\mathrm{In}\:\mathrm{triangle}\:\mathrm{ABC},\:\mathrm{AB}=\mathrm{c},\:\mathrm{BC}=\mathrm{b},\:\mathrm{AC}=\mathrm{a} \\ $$$$\mathrm{ab}^{\mathrm{2}} \mathrm{c}\:+\:\mathrm{abc}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \mathrm{bc}\:\geqslant\:\mathrm{tan}\:\frac{\mathrm{A}}{\mathrm{2}}\:\frac{\mathrm{2S}^{\mathrm{3}} }{\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} −\mathrm{c}^{\mathrm{2}} } \\ $$

Question Number 219110 Answers: 1 Comments: 0

find the nth term of x_(n+1) = x_n (2−x_n ) in x_1

$${find}\:{the}\:{nth}\:{term}\:{of}\:{x}_{{n}+\mathrm{1}} \:=\:{x}_{{n}} \left(\mathrm{2}−{x}_{{n}} \right) \\ $$$${in}\:{x}_{\mathrm{1}} \\ $$

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