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AlgebraQuestion and Answers: Page 62

Question Number 196872 Answers: 1 Comments: 0

Let ξ be a positive Root of x^2 −2023x−1 Define a sequence ϕ_i such That ϕ_0 =1 ϕ_(n+1) =⌊ϕ_n ξ⌋, find The Remainder When ϕ_(2023 ) is divided by (√ϕ_2 )

$${Let}\:\xi\:{be}\:{a}\:{positive}\:{Root}\:{of}\:{x}^{\mathrm{2}} −\mathrm{2023}{x}−\mathrm{1} \\ $$$${Define}\:{a}\:{sequence}\:\varphi_{{i}} \:{such}\:{That}\:\varphi_{\mathrm{0}} =\mathrm{1} \\ $$$$\varphi_{{n}+\mathrm{1}} =\lfloor\varphi_{{n}} \xi\rfloor,\:{find}\:{The}\:{Remainder}\:{When}\:\varphi_{\mathrm{2023}\:} {is}\:{divided}\:{by}\:\sqrt{\varphi_{\mathrm{2}} } \\ $$

Question Number 196870 Answers: 1 Comments: 0

let b_i ∧ a_i >0 where i∈{1,2,3,...,n}& Σ_(i=1) ^n (b_i )=λ Prove that ((λ−(b_1 +b_2 ))/((b_1 +b_2 )))(a_1 +a_2 )+((λ−(b_1 +b_3 ))/((b_1 +b_3 )))(a_1 +a_3 )+....+((λ−(b_2 +b_3 ))/((b_2 +b_3 )))(a_2 +a_3 )+...((λ−(b_(n−1) +b_n ))/((b_(n−1) +b_n )))(a_(n−1) +a_n ) ≥(√(((n(n−1)(n−2)^2 )/4)×ΣΣ_(1≤i<j≤n) (a_i a_j )))

$${let}\:{b}_{{i}} \wedge\:{a}_{{i}} >\mathrm{0}\:{where}\:{i}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},...,{n}\right\}\&\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left({b}_{{i}} \right)=\lambda\:{Prove}\:{that} \\ $$$$\frac{\lambda−\left({b}_{\mathrm{1}} +{b}_{\mathrm{2}} \right)}{\left({b}_{\mathrm{1}} +{b}_{\mathrm{2}} \right)}\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} \right)+\frac{\lambda−\left({b}_{\mathrm{1}} +{b}_{\mathrm{3}} \right)}{\left({b}_{\mathrm{1}} +{b}_{\mathrm{3}} \right)}\left({a}_{\mathrm{1}} +{a}_{\mathrm{3}} \right)+....+\frac{\lambda−\left({b}_{\mathrm{2}} +{b}_{\mathrm{3}} \right)}{\left({b}_{\mathrm{2}} +{b}_{\mathrm{3}} \right)}\left({a}_{\mathrm{2}} +{a}_{\mathrm{3}} \right)+...\frac{\lambda−\left({b}_{{n}−\mathrm{1}} +{b}_{{n}} \right)}{\left({b}_{{n}−\mathrm{1}} +{b}_{{n}} \right)}\left({a}_{{n}−\mathrm{1}} +{a}_{{n}} \right) \\ $$$$\geqslant\sqrt{\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{4}}×\underset{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} {\Sigma\Sigma}\left({a}_{{i}} {a}_{{j}} \right)} \\ $$$$ \\ $$

Question Number 196861 Answers: 0 Comments: 0

Question Number 196860 Answers: 0 Comments: 0

Question Number 196815 Answers: 1 Comments: 0

!6×(((!5+9!!!!!+7!!!−16))^(1/4) /(!10))=?

$$!\mathrm{6}×\frac{\sqrt[{\mathrm{4}}]{!\mathrm{5}+\mathrm{9}!!!!!+\mathrm{7}!!!−\mathrm{16}}}{!\mathrm{10}}=? \\ $$

Question Number 196728 Answers: 1 Comments: 1

Question Number 196730 Answers: 0 Comments: 0

Question Number 196714 Answers: 3 Comments: 0

Question Number 196704 Answers: 0 Comments: 0

Question Number 196693 Answers: 0 Comments: 0

And If I want to study an abstract algebra what book would you recommend and are there any prequesties

$$ \\ $$And If I want to study an abstract algebra what book would you recommend and are there any prequesties

Question Number 196667 Answers: 3 Comments: 0

Question Number 196663 Answers: 0 Comments: 0

Question Number 196639 Answers: 2 Comments: 0

if sinx=(2/3) then find you the value of sin^6 x+cos^6 x???

$$\mathrm{if}\:\mathrm{sinx}=\frac{\mathrm{2}}{\mathrm{3}}\:\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{sin}}^{\mathrm{6}} \boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}^{\mathrm{6}} \boldsymbol{\mathrm{x}}??? \\ $$

Question Number 196633 Answers: 0 Comments: 1

Question Number 196770 Answers: 1 Comments: 2

Question Number 196629 Answers: 1 Comments: 1

if xyz=1, prove ((x/(x−1)))^2 +((y/(y−1)))^2 +((z/(z−1)))^2 ≥1.

$${if}\:{xyz}=\mathrm{1},\:{prove} \\ $$$$\left(\frac{{x}}{{x}−\mathrm{1}}\right)^{\mathrm{2}} +\left(\frac{{y}}{{y}−\mathrm{1}}\right)^{\mathrm{2}} +\left(\frac{{z}}{{z}−\mathrm{1}}\right)^{\mathrm{2}} \geqslant\mathrm{1}. \\ $$

Question Number 196582 Answers: 1 Comments: 0

If x = log_a bc, y = log_b ca and z = log_c ab then prove that x + y + z = xyz − 2.

$$\mathrm{If}\:{x}\:=\:\mathrm{log}_{{a}} {bc},\:{y}\:=\:\mathrm{log}_{{b}} {ca}\:\mathrm{and}\:{z}\:=\:\mathrm{log}_{{c}} {ab} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:{x}\:+\:{y}\:+\:{z}\:=\:{xyz}\:−\:\mathrm{2}. \\ $$

Question Number 196555 Answers: 0 Comments: 0

In △ABC show that Σ ((1 + cos ∙ (A − B) ∙ cos C)/(h_C ∙ sec C)) = (3/(2 R))

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{show}\:\mathrm{that} \\ $$$$\Sigma\:\frac{\mathrm{1}\:+\:\mathrm{cos}\:\centerdot\:\left(\mathrm{A}\:−\:\mathrm{B}\right)\:\centerdot\:\mathrm{cos}\:\mathrm{C}}{\mathrm{h}_{\boldsymbol{\mathrm{C}}} \:\centerdot\:\mathrm{sec}\:\mathrm{C}}\:\:=\:\:\frac{\mathrm{3}}{\mathrm{2}\:\mathrm{R}} \\ $$

Question Number 196614 Answers: 2 Comments: 0

if S_n =(1/(1+5n))+(1/(2+5n))+(1/(3+5n))+...+(1/(6n)), find lim_(n→∞) S_n =?

$${if}\:{S}_{{n}} =\frac{\mathrm{1}}{\mathrm{1}+\mathrm{5}{n}}+\frac{\mathrm{1}}{\mathrm{2}+\mathrm{5}{n}}+\frac{\mathrm{1}}{\mathrm{3}+\mathrm{5}{n}}+...+\frac{\mathrm{1}}{\mathrm{6}{n}}, \\ $$$${find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{S}_{{n}} =? \\ $$

Question Number 196612 Answers: 0 Comments: 1

Question Number 196544 Answers: 1 Comments: 0

Question Number 196519 Answers: 1 Comments: 0

if xyz=1, prove ((x/(x−1)))^2 +((y/(y−1)))^2 +((z/(z−1)))^2 ≥1.

$${if}\:{xyz}=\mathrm{1},\:{prove} \\ $$$$\left(\frac{{x}}{{x}−\mathrm{1}}\right)^{\mathrm{2}} +\left(\frac{{y}}{{y}−\mathrm{1}}\right)^{\mathrm{2}} +\left(\frac{{z}}{{z}−\mathrm{1}}\right)^{\mathrm{2}} \geqslant\mathrm{1}. \\ $$

Question Number 196515 Answers: 1 Comments: 0

Find: ∫_0 ^( +∞) x^𝛑 e^(−x) dx = ?

$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:+\infty} \:\mathrm{x}^{\boldsymbol{\pi}} \:\mathrm{e}^{−\boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:=\:? \\ $$

Question Number 196483 Answers: 0 Comments: 0

Question Number 196450 Answers: 3 Comments: 0

one solution of the equation (x−a)(x−b)(x−c)(x−d) = 9 is x=2. If a,b,c,d are different integers then a+b+c+d =?

$$\mathrm{one}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\left(\mathrm{x}−\mathrm{a}\right)\left(\mathrm{x}−\mathrm{b}\right)\left(\mathrm{x}−\mathrm{c}\right)\left(\mathrm{x}−\mathrm{d}\right)\:=\:\mathrm{9}\: \\ $$$$\:\mathrm{is}\:\mathrm{x}=\mathrm{2}.\:\mathrm{If}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{different}\: \\ $$$$\:\mathrm{integers}\:\mathrm{then}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\:=?\: \\ $$

Question Number 196364 Answers: 1 Comments: 0

xp(x)=x^3 −2x^2 +x−a p(−1)=?

$${xp}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +{x}−{a} \\ $$$${p}\left(−\mathrm{1}\right)=? \\ $$

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