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Question Number 88471 Answers: 1 Comments: 0

Question Number 88461 Answers: 0 Comments: 2

Question Number 88458 Answers: 0 Comments: 0

Using the principle of mathematical induction to prove that a_1 , a_2 , ... , a_n , ((a_1 + a_2 + ... + a_n )/n) ≥ ((a_1 , a_2 , ... , a_n ))^(1/n)

$$\mathrm{Using}\:\mathrm{the}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{to}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\:\:\mathrm{a}_{\mathrm{1}} \:,\:\:\:\mathrm{a}_{\mathrm{2}} \:,\:\:...\:,\:\mathrm{a}_{\mathrm{n}} \:,\:\:\frac{\mathrm{a}_{\mathrm{1}} \:+\:\mathrm{a}_{\mathrm{2}} \:+\:...\:+\:\mathrm{a}_{\mathrm{n}} }{\mathrm{n}}\:\:\:\:\geqslant\:\:\:\sqrt[{\mathrm{n}}]{\mathrm{a}_{\mathrm{1}} \:,\:\:\mathrm{a}_{\mathrm{2}} \:,\:\:...\:,\:\mathrm{a}_{\mathrm{n}} } \\ $$

Question Number 88456 Answers: 1 Comments: 4

Question Number 88462 Answers: 0 Comments: 3

Question Number 88440 Answers: 2 Comments: 2

prove Σ_(k=1) ^∞ ((4^k −3^k )/(12^k ))=(1/6)

$${prove}\:\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{4}^{{k}} −\mathrm{3}^{{k}} }{\mathrm{12}^{{k}} }=\frac{\mathrm{1}}{\mathrm{6}} \\ $$

Question Number 88439 Answers: 0 Comments: 4

kofi have three books on his desk, they are mathematics , biology and physics. Ama also have three books on her desk namely physics ,mathematics and chemistry. A thief picked one book from each of their desk. what is the probability of picking mathematics book

$$\mathrm{kofi}\:\mathrm{have}\:\mathrm{three}\:\mathrm{books}\:\mathrm{on}\:\mathrm{his}\:\mathrm{desk},\:\mathrm{they} \\ $$$$\mathrm{are}\:\mathrm{mathematics}\:,\:\mathrm{biology}\:\mathrm{and}\:\mathrm{physics}. \\ $$$$\mathrm{Ama}\:\mathrm{also}\:\mathrm{have}\:\mathrm{three}\:\mathrm{books}\:\mathrm{on}\: \\ $$$$\mathrm{her}\:\mathrm{desk}\:\mathrm{namely}\:\mathrm{physics}\:,\mathrm{mathematics} \\ $$$$\mathrm{and}\:\mathrm{chemistry}.\:\mathrm{A}\:\mathrm{thief}\:\mathrm{picked}\:\mathrm{one}\:\mathrm{book} \\ $$$$\mathrm{from}\:\mathrm{each}\:\mathrm{of}\:\mathrm{their}\:\mathrm{desk}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{probability}\:\mathrm{of}\:\mathrm{picking}\:\mathrm{mathematics}\: \\ $$$$\mathrm{book} \\ $$

Question Number 88438 Answers: 2 Comments: 0

∫((x^5 +1)/(x^5 −1))dx

$$\int\frac{{x}^{\mathrm{5}} +\mathrm{1}}{{x}^{\mathrm{5}} −\mathrm{1}}{dx} \\ $$

Question Number 88436 Answers: 0 Comments: 0

show that (for e>1) the equation of a hyperbola with focus (±ae,0) and directrix x = (a/e) is (x^2 /a^2 ) − (y^2 /b^2 ) hence find an equation for the eccencitrity of the hyperbola

$$\:\mathrm{show}\:\mathrm{that}\:\left(\mathrm{for}\:{e}>\mathrm{1}\right)\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hyperbola}\:\mathrm{with}\: \\ $$$$\mathrm{focus}\:\:\left(\pm{ae},\mathrm{0}\right)\:\mathrm{and}\:\mathrm{directrix}\:\:{x}\:=\:\frac{{a}}{{e}}\:\mathrm{is}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:−\:\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} } \\ $$$$\:\:\:\mathrm{hence}\:\mathrm{find}\:\mathrm{an}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{eccencitrity}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{hyperbola} \\ $$

Question Number 88435 Answers: 0 Comments: 0

Question Number 88434 Answers: 0 Comments: 2

find the partial derivatives of first and second order for the function f(x,y) = x^3 y + 3xy + y^4

$$\mathrm{find}\:\mathrm{the}\:\mathrm{partial}\:\mathrm{derivatives}\:\mathrm{of}\:\mathrm{first}\:\mathrm{and}\:\mathrm{second}\:\mathrm{order} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{function} \\ $$$$\:{f}\left({x},{y}\right)\:=\:{x}^{\mathrm{3}} {y}\:+\:\mathrm{3}{xy}\:+\:{y}^{\mathrm{4}} \\ $$

Question Number 88430 Answers: 1 Comments: 1

Question Number 88429 Answers: 0 Comments: 0

show that ∫_(0 ) ^1 ln(x) sin^(−1) (√x) dx= (π/2)(ln(2)−1)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}\:} ^{\mathrm{1}} {ln}\left({x}\right)\:{sin}^{−\mathrm{1}} \sqrt{{x}}\:{dx}=\:\frac{\pi}{\mathrm{2}}\left({ln}\left(\mathrm{2}\right)−\mathrm{1}\right) \\ $$

Question Number 88424 Answers: 0 Comments: 1

calculate U_n =∫_0 ^∞ ((arctan(n^2 x)−arctan(nx))/x)dx and xetermine nature of the serie Σ U_n

$${calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({n}^{\mathrm{2}} {x}\right)−{arctan}\left({nx}\right)}{{x}}{dx} \\ $$$${and}\:{xetermine}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 88423 Answers: 0 Comments: 1

solve (x^2 −1)y^(′′) +(√x)y^′ =xe^(−x)

$${solve}\:\left({x}^{\mathrm{2}} −\mathrm{1}\right){y}^{''} \:+\sqrt{{x}}{y}^{'} \:={xe}^{−{x}} \\ $$

Question Number 88422 Answers: 0 Comments: 1

calculate ∫_1 ^∞ (dx/((x+1)^3 (x^2 +1)^2 ))

$${calculate}\:\:\int_{\mathrm{1}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 88419 Answers: 0 Comments: 0

find L(((arctanx)/x))

$${find}\:{L}\left(\frac{{arctanx}}{{x}}\right) \\ $$

Question Number 88417 Answers: 0 Comments: 1

e^(−z) _1 f_1 (a;b;z)=((Γ(b))/(Γ(b−a))) G_(1,2) ^(1,1) (z∣_(0,1−b) ^(a−b+1) )

$${e}^{−{z}} \:_{\mathrm{1}} {f}_{\mathrm{1}} \left({a};{b};{z}\right)=\frac{\Gamma\left({b}\right)}{\Gamma\left({b}−{a}\right)}\:{G}_{\mathrm{1},\mathrm{2}} ^{\mathrm{1},\mathrm{1}} \left({z}\mid_{\mathrm{0},\mathrm{1}−{b}} ^{{a}−{b}+\mathrm{1}} \right) \\ $$

Question Number 88415 Answers: 0 Comments: 2

find L(((1−cosx)/x^2 )) with L lsplace transform

$${find}\:{L}\left(\frac{\mathrm{1}−{cosx}}{{x}^{\mathrm{2}} }\right)\:{with}\:{L}\:{lsplace}\:{transform} \\ $$

Question Number 88414 Answers: 0 Comments: 2

find approcimstive value of ∫_(π/3) ^(π/2) (x/(sinx))dx

$${find}\:{approcimstive}\:{value}\:{of}\:\:\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{x}}{{sinx}}{dx} \\ $$

Question Number 88413 Answers: 0 Comments: 3

∫_0 ^∞ e^(−x^2 ) dx

$$\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$$$ \\ $$

Question Number 88388 Answers: 1 Comments: 0

Question Number 88385 Answers: 0 Comments: 0

if f(x)=(√(x−2)) is there cirtical point in (2,0)

$${if}\:\:\:{f}\left({x}\right)=\sqrt{{x}−\mathrm{2}} \\ $$$${is}\:{there}\:{cirtical}\:{point}\:{in}\:\left(\mathrm{2},\mathrm{0}\right)\: \\ $$$$ \\ $$

Question Number 88378 Answers: 1 Comments: 4

find the equation of a parabola with focus (3,3) and directrix y = 0

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{parabola}\:\mathrm{with}\:\mathrm{focus}\:\left(\mathrm{3},\mathrm{3}\right) \\ $$$$\mathrm{and}\:\mathrm{directrix}\:\:{y}\:=\:\mathrm{0} \\ $$

Question Number 88372 Answers: 0 Comments: 10

There are four boxes, each of them contains exactly the same numbers: 1,2,3,...,n. Four different numbers are drawn from the boxes and multiplicated with each other to get a product. What′s the sum of all products? Σ_(a≠b≠c≠d) abcd=?

$${There}\:{are}\:{four}\:{boxes},\:{each}\:{of}\:{them} \\ $$$${contains}\:{exactly}\:{the}\:{same}\:{numbers}: \\ $$$$\mathrm{1},\mathrm{2},\mathrm{3},...,{n}. \\ $$$${Four}\:{different}\:{numbers}\:{are}\:{drawn} \\ $$$${from}\:{the}\:{boxes}\:{and}\:{multiplicated} \\ $$$${with}\:{each}\:{other}\:{to}\:{get}\:{a}\:{product}. \\ $$$${What}'{s}\:{the}\:{sum}\:{of}\:{all}\:{products}? \\ $$$$\underset{{a}\neq{b}\neq{c}\neq{d}} {\sum}{abcd}=? \\ $$

Question Number 88364 Answers: 1 Comments: 0

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