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Question Number 147585    Answers: 1   Comments: 0

Question Number 147582    Answers: 1   Comments: 0

if x;y;z>0 prove that ((x^3 +y^3 +z^3 )/(xyz)) ≥ 2((x/(y+z)) + (y/(z+x)) + (z/(x+y)))

$${if}\:\:{x};{y};{z}>\mathrm{0}\:\:{prove}\:{that} \\ $$$$\frac{{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} }{{xyz}}\:\geqslant\:\mathrm{2}\left(\frac{{x}}{{y}+{z}}\:+\:\frac{{y}}{{z}+{x}}\:+\:\frac{{z}}{{x}+{y}}\right) \\ $$

Question Number 147581    Answers: 1   Comments: 0

2sin 2x −4sin^2 x = 7cos 2x (π/2)<x<π ⇒ sin 2x =?

$$\:\:\mathrm{2sin}\:\mathrm{2x}\:−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}\:=\:\mathrm{7cos}\:\mathrm{2x}\: \\ $$$$\:\frac{\pi}{\mathrm{2}}<\mathrm{x}<\pi\:\Rightarrow\:\mathrm{sin}\:\mathrm{2x}\:=? \\ $$

Question Number 147576    Answers: 1   Comments: 0

(1):: Σ_(i=1) ^n Σ_(j=1) ^n ∣i−j∣=? (2):: Σ_(i=1) ^n Σ_(j=i) ^n (1/j)=? (3):: Σ_(i=1) ^n^2 [(√i)]=?

$$\left(\mathrm{1}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mid\mathrm{i}−\mathrm{j}\mid=? \\ $$$$\left(\mathrm{2}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{i}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{j}}=? \\ $$$$\left(\mathrm{3}\right)::\:\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}^{\mathrm{2}} } {\sum}}\left[\sqrt{\mathrm{i}}\right]=? \\ $$

Question Number 147573    Answers: 0   Comments: 1

Question Number 147572    Answers: 2   Comments: 0

Σ_(n≥1) ((4n−3)/((n^2 +2n)(n+3))) =?

$$\:\:\:\underset{{n}\geqslant\mathrm{1}} {\sum}\:\frac{\mathrm{4}{n}−\mathrm{3}}{\left({n}^{\mathrm{2}} +\mathrm{2}{n}\right)\left({n}+\mathrm{3}\right)}\:=? \\ $$

Question Number 147569    Answers: 1   Comments: 0

find the taylor series of f(z)=sinz ,z=(π/4) in complex number

$${find}\:{the}\:{taylor}\:{series}\:{of}\:{f}\left({z}\right)={sinz}\:,{z}=\frac{\pi}{\mathrm{4}}\:{in}\:{complex}\:{number} \\ $$

Question Number 147566    Answers: 1   Comments: 0

x^3 + 3367 = 2^n ⇒ x ; n = ?

$${x}^{\mathrm{3}} \:+\:\mathrm{3367}\:=\:\mathrm{2}^{\boldsymbol{{n}}} \:\:\Rightarrow\:{x}\:;\:{n}\:=\:? \\ $$

Question Number 147561    Answers: 1   Comments: 1

Question Number 147557    Answers: 2   Comments: 0

Simlify (((1+(√x))/( (√(1+x)))) − ((√(1+x))/(1+(√x))))^2 - (((1−(√x))/( (√(1+x)))) − ((√(1+x))/(1−(√x))))^2

$${Simlify} \\ $$$$\left(\frac{\mathrm{1}+\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}}}\:−\:\frac{\sqrt{\mathrm{1}+{x}}}{\mathrm{1}+\sqrt{{x}}}\right)^{\mathrm{2}} -\:\left(\frac{\mathrm{1}−\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}}}\:−\:\frac{\sqrt{\mathrm{1}+{x}}}{\mathrm{1}−\sqrt{{x}}}\right)^{\mathrm{2}} \\ $$

Question Number 147554    Answers: 1   Comments: 0

A toroid core has N=1200 turns, length L=80cm,cross-section area A=60cm^2 ,current I=1.5A. Compute B and H.Assume an empty core

$$\mathrm{A}\:\mathrm{toroid}\:\mathrm{core}\:\mathrm{has}\:\mathrm{N}=\mathrm{1200}\:\mathrm{turns}, \\ $$$$\mathrm{length}\:\mathrm{L}=\mathrm{80cm},\mathrm{cross}-\mathrm{section}\:\mathrm{area} \\ $$$$\mathrm{A}=\mathrm{60cm}^{\mathrm{2}} ,\mathrm{current}\:\mathrm{I}=\mathrm{1}.\mathrm{5A}. \\ $$$$\:\:\mathrm{Compute}\:\mathrm{B}\:\mathrm{and}\:\mathrm{H}.\mathrm{Assume}\:\mathrm{an} \\ $$$$\:\mathrm{empty}\:\mathrm{core} \\ $$

Question Number 147553    Answers: 1   Comments: 0

Question Number 147543    Answers: 2   Comments: 0

Π_(m=1) ^n ((1/2))^m

$$\underset{{m}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{m}} \\ $$

Question Number 147539    Answers: 2   Comments: 0

show that (1/2)∫_1 ^( 2) (1/x^5 )dx ≤ ∫_1 ^( 2) (1/(x^4 +1))dx ≤∫_1 ^( 2) (1/x^4 )

$${show}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{5}} }{dx}\:\leqslant\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{1}}{dx}\:\leqslant\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$

Question Number 147535    Answers: 1   Comments: 1

Question Number 147529    Answers: 3   Comments: 1

if a_(n+1) = (√(a_1 + a_n )) find lim_(x→∞) a_n = ?

$${if}\:\:\:{a}_{\boldsymbol{{n}}+\mathrm{1}} \:=\:\sqrt{{a}_{\mathrm{1}} \:+\:{a}_{\boldsymbol{{n}}} } \\ $$$${find}\:\:\underset{\boldsymbol{{x}}\rightarrow\infty} {{lim}a}_{\boldsymbol{{n}}} \:=\:? \\ $$

Question Number 147528    Answers: 0   Comments: 3

q_n =Π_(n=0) ^∞ cos((x/2^n )) i) study the variation of q_n ii) show that cosx=((sin2x)/(2sinx)) , ∀x∈[0,(π/2)] iii) deduce that q_n =(1/2^(n+1) )×((sin2x)/(sin((x/2^n )))) iv)lim_(n→∞) q_n =? v) solve cos((x/2))≥−(1/2)

$${q}_{{n}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}{cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right) \\ $$$$\left.{i}\right)\:{study}\:{the}\:{variation}\:{of}\:{q}_{{n}} \\ $$$$\left.{ii}\right) \\ $$$$\:{show}\:{that}\:{cosx}=\frac{{sin}\mathrm{2}{x}}{\mathrm{2}{sinx}}\:,\:\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right] \\ $$$$\left.{iii}\right) \\ $$$${deduce}\:{that}\:{q}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}^{{n}+\mathrm{1}} }×\frac{{sin}\mathrm{2}{x}}{{sin}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)} \\ $$$$\left.{iv}\right)\underset{{n}\rightarrow\infty} {\mathrm{lim}}{q}_{{n}} =? \\ $$$$\left.{v}\right)\: \\ $$$${solve}\:{cos}\left(\frac{{x}}{\mathrm{2}}\right)\geqslant−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 147524    Answers: 0   Comments: 5

Question Number 147513    Answers: 1   Comments: 1

Question Number 147661    Answers: 1   Comments: 0

41^3 + 42^3 + 43^3 + ... + 59^3 Find the last three digits of the number

$$\mathrm{41}^{\mathrm{3}} \:+\:\mathrm{42}^{\mathrm{3}} \:+\:\mathrm{43}^{\mathrm{3}} \:+\:...\:+\:\mathrm{59}^{\mathrm{3}} \\ $$$${Find}\:{the}\:{last}\:{three}\:{digits}\:{of}\:{the} \\ $$$${number} \\ $$

Question Number 147507    Answers: 1   Comments: 2

Question Number 147697    Answers: 1   Comments: 0

f(0)+f(1)+f(2)+...+f(n)=n!−n∙a f(16)=15∙(15!−1) find a=?

$${f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+...+{f}\left({n}\right)={n}!−{n}\centerdot{a} \\ $$$${f}\left(\mathrm{16}\right)=\mathrm{15}\centerdot\left(\mathrm{15}!−\mathrm{1}\right) \\ $$$${find}\:\:\boldsymbol{{a}}=? \\ $$

Question Number 147500    Answers: 1   Comments: 1

x^2 - y^2 + 2x = 22 what is the number of complete solutions that satisf the equation (x;y).?

$${x}^{\mathrm{2}} \:-\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:=\:\mathrm{22} \\ $$$${what}\:{is}\:{the}\:{number}\:{of}\:{complete} \\ $$$${solutions}\:{that}\:{satisf}\:{the}\:{equation} \\ $$$$\left({x};{y}\right).? \\ $$

Question Number 147493    Answers: 0   Comments: 0

hi, dears masters ! A = {au+bv, (a, b, u, v) ∈ Z^4 } with a ≠ b. 1. prove that A is ideal of Z. 2. let 𝛌Z = {𝛌n , n ∈ Z}. prove that A has a smaller element 𝛌 strictly positive such that A = 𝛌Z. 3. prove that 𝛌 = gcd(a,b).

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{dears}}\:\boldsymbol{\mathrm{masters}}\:! \\ $$$$\boldsymbol{\mathrm{A}}\:=\:\left\{\boldsymbol{{au}}+\boldsymbol{{bv}},\:\left(\boldsymbol{{a}},\:\boldsymbol{{b}},\:\boldsymbol{{u}},\:\boldsymbol{{v}}\right)\:\in\:\mathbb{Z}^{\mathrm{4}} \right\}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{{a}}\:\neq\:\boldsymbol{{b}}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{ideal}}\:\boldsymbol{\mathrm{of}}\:\:\mathbb{Z}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{let}}\:\boldsymbol{\lambda}\mathbb{Z}\:=\:\left\{\boldsymbol{\lambda{n}}\:,\:\boldsymbol{{n}}\:\in\:\mathbb{Z}\right\}.\: \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{has}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{smaller}}\:\boldsymbol{\mathrm{element}}\:\boldsymbol{\lambda}\:\boldsymbol{\mathrm{strictly}}\: \\ $$$$\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{A}}\:=\:\boldsymbol{\lambda}\mathbb{Z}. \\ $$$$\mathrm{3}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\lambda}\:=\:\boldsymbol{\mathrm{gcd}}\left(\boldsymbol{{a}},\boldsymbol{{b}}\right). \\ $$

Question Number 147492    Answers: 2   Comments: 3

2(√(19)) cos [(1/3)tan^(−1) (((45(√3))/(28)))] it is equal to 8. How?

$$\mathrm{2}\sqrt{\mathrm{19}}\:\mathrm{cos}\:\left[\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{45}\sqrt{\mathrm{3}}}{\mathrm{28}}\right)\right] \\ $$$${it}\:{is}\:{equal}\:{to}\:\mathrm{8}.\:{How}? \\ $$

Question Number 147488    Answers: 1   Comments: 0

if x;y;z≥1 then: (1/(3xy−1)) + (1/(3yz−1)) + (1/(3zx−1)) ≥ (3/(2xyz))

$${if}\:\:{x};{y};{z}\geqslant\mathrm{1}\:\:{then}: \\ $$$$\frac{\mathrm{1}}{\mathrm{3}{xy}−\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{yz}−\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{zx}−\mathrm{1}}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}{xyz}} \\ $$

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