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Question Number 124107 Answers: 0 Comments: 0

n ∈ N and p is a prime number (p≥3). a and b are defined by: a=2^n and b=a×b. S(a) is the sum of divisors of a and S(b) is the sum of divisors of b. 1.Determinate the set of divisors of a and the set of divisors of b. 2.show that S(a)+2^(n+1) =1+2S(a) then calculate S(a). 3. write S(b) in function of S(a) then calculate S(b).

$${n}\:\in\:\mathbb{N}\:{and}\:{p}\:{is}\:{a}\:{prime}\:{number}\:\left({p}\geqslant\mathrm{3}\right). \\ $$$${a}\:{and}\:{b}\:{are}\:{defined}\:{by}:\:{a}=\mathrm{2}^{{n}} \:{and} \\ $$$${b}={a}×{b}.\:\:{S}\left({a}\right)\:{is}\:{the}\:{sum}\:{of}\:{divisors}\: \\ $$$${of}\:{a}\:{and}\:{S}\left({b}\right)\:\:{is}\:{the}\:{sum}\:{of}\:{divisors} \\ $$$${of}\:{b}. \\ $$$$\mathrm{1}.{Determinate}\:{the}\:{set}\:{of}\:{divisors}\:{of} \\ $$$${a}\:{and}\:{the}\:{set}\:{of}\:{divisors}\:{of}\:{b}. \\ $$$$\mathrm{2}.{show}\:{that}\:{S}\left({a}\right)+\mathrm{2}^{{n}+\mathrm{1}} =\mathrm{1}+\mathrm{2}{S}\left({a}\right) \\ $$$${then}\:{calculate}\:{S}\left({a}\right). \\ $$$$\mathrm{3}.\:{write}\:{S}\left({b}\right)\:{in}\:{function}\:{of}\:{S}\left({a}\right)\:{then} \\ $$$${calculate}\:{S}\left({b}\right). \\ $$

Question Number 124106 Answers: 0 Comments: 0

U_(n ) is a sequence of real numbers defined by U_0 =0 and for n ∈ N, U_(n+1) =(√(U_n +6)) 1. show that 0≤U_n ≤3. 2. show that U_(n ) is non−decreasing. 3. show that 3−U_(n+1) ≤((3−U_n )/3) 4. Deduct that 0≤3−U_(n+1) ≤((1/3))^n

$${U}_{{n}\:} {is}\:{a}\:{sequence}\:{of}\:{real}\:{numbers}\: \\ $$$${defined}\:{by}\:{U}_{\mathrm{0}} =\mathrm{0}\:{and}\:{for}\:{n}\:\in\:\mathbb{N},\: \\ $$$${U}_{{n}+\mathrm{1}} =\sqrt{{U}_{{n}} +\mathrm{6}} \\ $$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{0}\leqslant{U}_{{n}} \leqslant\mathrm{3}. \\ $$$$\mathrm{2}.\:{show}\:{that}\:{U}_{{n}\:} {is}\:{non}−{decreasing}. \\ $$$$\mathrm{3}.\:{show}\:{that}\:\mathrm{3}−{U}_{{n}+\mathrm{1}} \leqslant\frac{\mathrm{3}−{U}_{{n}} }{\mathrm{3}} \\ $$$$\mathrm{4}.\:\:{Deduct}\:{that}\:\mathrm{0}\leqslant\mathrm{3}−{U}_{{n}+\mathrm{1}} \leqslant\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{{n}} \\ $$

Question Number 124056 Answers: 0 Comments: 2

If α,β,γ are the real roots of the equation a^3 −6a^2 +3a+1=0 then find all possible values of the?expression α^2 β+β^2 γ+γ^2 α

$${If}\:\alpha,\beta,\gamma\:{are}\:{the}\:{real}\:{roots}\:{of}\:{the}\:{equation}\: \\ $$$${a}^{\mathrm{3}} −\mathrm{6}{a}^{\mathrm{2}} +\mathrm{3}{a}+\mathrm{1}=\mathrm{0}\:\:\:\:\:{then}\:{find}\:{all}\:{possible}\:{values} \\ $$$${of}\:{the}?{expression}\:\alpha^{\mathrm{2}} \beta+\beta^{\mathrm{2}} \gamma+\gamma^{\mathrm{2}} \alpha \\ $$

Question Number 123987 Answers: 1 Comments: 0

...nice mathematics... if f(x)=x^3 +x then find f^( −1) (x)=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{mathematics}... \\ $$$$\:\:\:{if}\:\:\:\:{f}\left({x}\right)={x}^{\mathrm{3}} +{x}\:\:{then} \\ $$$$\:\:\:\:\:\:\:\:\:{find}\:\:\:{f}^{\:−\mathrm{1}} \left({x}\right)=? \\ $$$$\:\:\:\: \\ $$

Question Number 123894 Answers: 2 Comments: 0

Question Number 123886 Answers: 1 Comments: 0

Let l^− z+lz^− +m=0 be a straight line in the complex plane and P(z_0 ) be a point in the plane. Then the equation of the line passing through P(z_0 ) and perpendicular to the given line is ___

$$\mathrm{Let}\:\overset{−} {{l}z}+{l}\overset{−} {{z}}+{m}=\mathrm{0}\:\mathrm{be}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{in}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane} \\ $$$$\mathrm{and}\:{P}\left({z}_{\mathrm{0}} \right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{plane}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{passing}\:\mathrm{through}\:{P}\left({z}_{\mathrm{0}} \right)\:\mathrm{and}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{given}\:\mathrm{line}\:\mathrm{is}\:\_\_\_ \\ $$

Question Number 123824 Answers: 1 Comments: 1

Show that for all real numbers (x/y/z) satisfying x+y+z=0 and xy +yz+zx=−3 the value of expression x^3 y+y^3 z +z^3 x is a constant

$${Show}\:{that}\:{for}\:{all}\:{real}\:{numbers}\:\left({x}/{y}/{z}\right)\:{satisfying}\:\: \\ $$$${x}+{y}+{z}=\mathrm{0}\:{and}\:{xy}\:+{yz}+{zx}=−\mathrm{3}\:\:\:{the}\:{value}\:{of} \\ $$$${expression}\:{x}^{\mathrm{3}} {y}+{y}^{\mathrm{3}} {z}\:+{z}^{\mathrm{3}} {x}\:\:\:{is}\:{a}\:{constant} \\ $$

Question Number 123746 Answers: 0 Comments: 1

Question Number 123638 Answers: 0 Comments: 1

show that if x∈R then x and −x cannot be positive

$$\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{x}\in\mathbb{R}\:\mathrm{then}\:\mathrm{x}\:\mathrm{and}\:−\mathrm{x}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{positive} \\ $$

Question Number 123626 Answers: 0 Comments: 0

Question Number 123595 Answers: 0 Comments: 1

For real numbers (a/b/c) define s_n =a^n +b^n +c^n Suppose s_1 =2 s_2 =6 and s_(3 ) =14.Prove that ∣s_n ^2 −s_(n−1) s_(n+1) ∣=8=holds ∀ n>1

$${For}\:{real}\:{numbers}\:\left({a}/{b}/{c}\right)\:{define}\:{s}_{{n}} ={a}^{{n}} +{b}^{{n}} +{c}^{{n}} \\ $$$${Suppose}\:{s}_{\mathrm{1}} =\mathrm{2}\:{s}_{\mathrm{2}} =\mathrm{6}\:{and}\:{s}_{\mathrm{3}\:} =\mathrm{14}.{Prove}\:{that}\: \\ $$$$\mid{s}_{{n}} ^{\mathrm{2}} −{s}_{{n}−\mathrm{1}} {s}_{{n}+\mathrm{1}} \mid=\mathrm{8}={holds}\:\forall\:{n}>\mathrm{1} \\ $$

Question Number 123591 Answers: 0 Comments: 2

Question Number 123576 Answers: 1 Comments: 2

Question Number 123574 Answers: 1 Comments: 0

Question Number 123548 Answers: 1 Comments: 1

a man 6 ft tall walks at a rate of 7 ft/sec away from a lamppost that is 14 ft hight. at what rate is the lenght of his shadow charging when he is 30 ft away from the lamppost

$${a}\:{man}\:\mathrm{6}\:{ft}\:{tall}\:{walks}\:{at}\:{a}\:{rate} \\ $$$${of}\:\mathrm{7}\:{ft}/{sec}\:{away}\:{from}\:{a}\:{lamppost} \\ $$$${that}\:{is}\:\mathrm{14}\:{ft}\:{hight}. \\ $$$${at}\:{what}\:{rate}\:{is}\:{the}\:{lenght}\:{of}\:{his} \\ $$$${shadow}\:{charging}\:{when}\:{he}\:{is}\:\mathrm{30}\:{ft}\: \\ $$$${away}\:{from}\:{the}\:{lamppost} \\ $$

Question Number 123510 Answers: 0 Comments: 0

sorry −(π/2)

$${sorry}\:−\frac{\pi}{\mathrm{2}} \\ $$

Question Number 123509 Answers: 1 Comments: 0

∫_(−(π/)2) ^(π/2) ∣cosx∣=?

$$\underset{−\frac{\pi}{}\mathrm{2}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mid{cosx}\mid=? \\ $$

Question Number 123426 Answers: 0 Comments: 5

find the radii of all circles with center C tangenting the curve h C= ((5),(1) ) h: x^2 −y^2 =1

$$\mathrm{find}\:\mathrm{the}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{all}\:\mathrm{circles}\:\mathrm{with}\:\mathrm{center}\:{C} \\ $$$$\mathrm{tangenting}\:\mathrm{the}\:\mathrm{curve}\:{h} \\ $$$${C}=\begin{pmatrix}{\mathrm{5}}\\{\mathrm{1}}\end{pmatrix} \\ $$$${h}:\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{1} \\ $$

Question Number 123409 Answers: 1 Comments: 0

x^4 +12x−5=0 show that the sum of two roots of the equation is 2