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Question Number 194638 Answers: 1 Comments: 1

Prove that ∀n∈IN^∗ Σ_(k=1) ^(2^n −1) (1/(sin^2 (((kπ)/2^(n+1) ))))= ((2^(2n+1) −2)/3) Give in terms of n Σ_(k=1) ^(2^n −1) (1/(sin^4 (((kπ)/2^(n+1) ))))

$$\mathrm{Prove}\:\mathrm{that}\:\forall{n}\in\mathrm{IN}^{\ast} \:\:\:\:\: \\ $$$$\:\:\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}^{{n}} −\mathrm{1}} {\sum}}\:\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\frac{{k}\pi}{\mathrm{2}^{{n}+\mathrm{1}} }\right)}=\:\frac{\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} −\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{Give}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}\:\:\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}^{{n}} −\mathrm{1}} {\sum}}\:\frac{\mathrm{1}}{{sin}^{\mathrm{4}} \left(\frac{{k}\pi}{\mathrm{2}^{{n}+\mathrm{1}} }\right)} \\ $$

Question Number 194637 Answers: 4 Comments: 1

x+y=1 x^2 +y^2 =2 x^(11) +y^(11) =?

$$ \\ $$$${x}+{y}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{\mathrm{11}} +{y}^{\mathrm{11}} =? \\ $$$$ \\ $$$$ \\ $$

Question Number 194636 Answers: 0 Comments: 3

Question Number 194634 Answers: 1 Comments: 0

a_1 ,a_2 ,a_3 ,....,a_n >0 such that a_i ∈[0,i] ∀ i∈{1,2,3,4,...,n} prove that 2^n .a_1 (a_1 +a_2 )...(a_1 +a_2 +...+a_n )≥(n+1)(a_1 ^2 .a_2 ^2 ...a_n ^2 )

$${a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ,{a}_{\mathrm{3}} ,....,{a}_{{n}} >\mathrm{0}\:{such}\:{that}\:{a}_{{i}} \in\left[\mathrm{0},{i}\right]\: \\ $$$$\forall\:{i}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},...,{n}\right\}\:{prove}\:{that} \\ $$$$\mathrm{2}^{{n}} .{a}_{\mathrm{1}} \left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} \right)...\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{{n}} \right)\geqslant\left({n}+\mathrm{1}\right)\left({a}_{\mathrm{1}} ^{\mathrm{2}} .{a}_{\mathrm{2}} ^{\mathrm{2}} ...{a}_{{n}} ^{\mathrm{2}} \right) \\ $$

Question Number 194624 Answers: 2 Comments: 2

Question Number 194619 Answers: 1 Comments: 0

Find the sum of the roots of the equation: −3x^3 + 8x^2 − 6x − 7 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}: \\ $$$$−\mathrm{3x}^{\mathrm{3}} \:+\:\mathrm{8x}^{\mathrm{2}} \:−\:\mathrm{6x}\:−\:\mathrm{7}\:=\:\mathrm{0} \\ $$

Question Number 194612 Answers: 1 Comments: 2

Question Number 194613 Answers: 2 Comments: 0

log _(4x) (x)+ log _(x/2) (x)= 2

$$\:\:\:\:\: \\ $$$$\:\:\mathrm{log}\:_{\mathrm{4x}} \left(\mathrm{x}\right)+\:\mathrm{log}\:_{\mathrm{x}/\mathrm{2}} \left(\mathrm{x}\right)=\:\mathrm{2}\: \\ $$

Question Number 194610 Answers: 1 Comments: 0

where can I learn about multiple sigma notaions of dependent and independent variables something like this Σ_(1≤i) Σ_(<j) Σ_(<k≤1) (i+j+k)=λ find λ I want to know what to study

$${where}\:{can}\:{I}\:{learn}\:{about}\:{multiple}\:{sigma}\:{notaions} \\ $$$${of}\:{dependent}\:{and}\:{independent}\:{variables} \\ $$$$ \\ $$$${something}\:{like}\:{this} \\ $$$$\underset{\mathrm{1}\leqslant{i}} {\sum}\underset{<{j}} {\sum}\underset{<{k}\leqslant\mathrm{1}} {\sum}\left({i}+{j}+{k}\right)=\lambda \\ $$$${find}\:\lambda \\ $$$${I}\:{want}\:{to}\:{know}\:{what}\:{to}\:{study} \\ $$

Question Number 194606 Answers: 0 Comments: 0

When a kichen is removed from an oven, its temperature is measured at 300^0 F. Three minutes later, its temperature is 200^0 F. How longwill it take the kitchen to cool of to a room temperature of 70^0 F?

$$\mathrm{When}\:\mathrm{a}\:\mathrm{kichen}\:\mathrm{is}\:\mathrm{removed}\:\mathrm{from}\:\mathrm{an} \\ $$$$\:\mathrm{oven},\:\mathrm{its}\:\mathrm{temperature}\:\mathrm{is}\:\mathrm{measured}\:\mathrm{at} \\ $$$$\:\mathrm{300}^{\mathrm{0}} \mathrm{F}.\:\mathrm{Three}\:\mathrm{minutes}\:\mathrm{later},\:\mathrm{its} \\ $$$$\:\mathrm{temperature}\:\mathrm{is}\:\mathrm{200}^{\mathrm{0}} \mathrm{F}.\:\mathrm{How}\:\mathrm{longwill} \\ $$$$\:\mathrm{it}\:\mathrm{take}\:\mathrm{the}\:\mathrm{kitchen}\:\mathrm{to}\:\mathrm{cool}\:\mathrm{of}\:\mathrm{to}\:\mathrm{a}\: \\ $$$$\mathrm{room}\:\mathrm{temperature}\:\mathrm{of}\:\mathrm{70}^{\mathrm{0}} \mathrm{F}? \\ $$

Question Number 194604 Answers: 0 Comments: 0

Question Number 194602 Answers: 1 Comments: 0

A ball is thrown vertically upwards with a velocity of 10m/s from a point 75 m above the ground. Calculate the velocity with which it hits the ground.

$$ \\ $$A ball is thrown vertically upwards with a velocity of 10m/s from a point 75 m above the ground. Calculate the velocity with which it hits the ground.

Question Number 194600 Answers: 1 Comments: 0

An object is pulled up a smooth plane inclined at angle 45° to the horizontal. If the plane is 25 m long and the object comes to rest at the top, correct to two decimal places the: (i) initial speed of the object (ii) time taken to reach the top.

$$ \\ $$An object is pulled up a smooth plane inclined at angle 45° to the horizontal. If the plane is 25 m long and the object comes to rest at the top, correct to two decimal places the: (i) initial speed of the object (ii) time taken to reach the top.

Question Number 194599 Answers: 0 Comments: 0

A tank contains 300 litres of fluid in which 20 grams of salt is dissolved. Brine containing 1 gm of salt per litre is then pumped into the tank at a rate of 4L/min; the well mixed solution is pumped out at the same rate. Find the number N(t) of grams of salt in the tank at time t.

$$\mathrm{A}\:\mathrm{tank}\:\mathrm{contains}\:\mathrm{300}\:\mathrm{litres}\:\mathrm{of}\:\mathrm{fluid}\:\mathrm{in}\: \\ $$$$\mathrm{which}\:\mathrm{20}\:\mathrm{grams}\:\mathrm{of}\:\mathrm{salt}\:\mathrm{is}\:\mathrm{dissolved}.\: \\ $$$$\mathrm{Brine}\:\mathrm{containing}\:\:\mathrm{1}\:\mathrm{gm}\:\mathrm{of}\:\mathrm{salt}\:\mathrm{per}\:\mathrm{litre} \\ $$$$\:\mathrm{is}\:\mathrm{then}\:\mathrm{pumped}\:\mathrm{into}\:\mathrm{the}\:\mathrm{tank}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate} \\ $$$$\:\mathrm{of}\:\mathrm{4L}/\mathrm{min};\:\mathrm{the}\:\mathrm{well}\:\mathrm{mixed}\:\mathrm{solution}\: \\ $$$$\mathrm{is}\:\mathrm{pumped}\:\mathrm{out}\:\mathrm{at}\:\mathrm{the}\:\mathrm{same}\:\mathrm{rate}. \\ $$$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{N}\left(\mathrm{t}\right)\:\mathrm{of}\:\mathrm{grams}\:\mathrm{of} \\ $$$$\:\mathrm{salt}\:\mathrm{in}\:\mathrm{the}\:\mathrm{tank}\:\mathrm{at}\:\mathrm{time}\:\mathrm{t}.\: \\ $$

Question Number 194598 Answers: 0 Comments: 2

please what is the best android apps to drow complex functions and fractals ?

$${please} \\ $$$${what}\:{is}\:{the}\:{best}\:{android} \\ $$$${apps}\:{to}\:{drow}\:{complex}\: \\ $$$${functions}\:{and}\:{fractals}\:? \\ $$

Question Number 194593 Answers: 3 Comments: 0

Question Number 194591 Answers: 0 Comments: 0

∫_(−2) ^2 ∫_(2x^2 ) ^8 ∫_(−(√((1/2)y−x^2 ))) ^(√((1/2)y−x^2 )) ((√(3x^2 +3z^2 )) )dzdydx

$$\:\:\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\:\underset{\mathrm{2x}^{\mathrm{2}} } {\overset{\mathrm{8}} {\int}}\:\:\underset{−\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}−\mathrm{x}^{\mathrm{2}} }} {\overset{\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}−\mathrm{x}^{\mathrm{2}} }} {\int}}\:\left(\sqrt{\mathrm{3x}^{\mathrm{2}} +\mathrm{3z}^{\mathrm{2}} }\:\right)\mathrm{dzdydx} \\ $$

Question Number 194586 Answers: 1 Comments: 2

abc = e^3 + d^3 + f^3 edf = a^3 + b^3 + c^3 find: abc and edf

$$\mathrm{abc}\:=\:\mathrm{e}^{\mathrm{3}} \:+\:\mathrm{d}^{\mathrm{3}} \:+\:\mathrm{f}^{\mathrm{3}} \\ $$$$\mathrm{edf}\:=\:\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \\ $$$$\mathrm{find}:\:\mathrm{abc}\:\:\mathrm{and}\:\:\mathrm{edf}\: \\ $$

Question Number 194579 Answers: 2 Comments: 0

if u_n =(1/( (√5)))[(((1+(√5))/2))^n −(((1−(√5))/2))^n ] then u_(n+1) =u_n +u_(n−1) ? ; n=0,1,2,..

$${if}\:\:\:\:{u}_{{n}} =\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\left[\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}} −\left(\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}} \right] \\ $$$$\:{then}\:\:\:{u}_{{n}+\mathrm{1}} ={u}_{{n}} +{u}_{{n}−\mathrm{1}} \:\:\:?\:\:\:\:\:;\:\:\:{n}=\mathrm{0},\mathrm{1},\mathrm{2},.. \\ $$

Question Number 194573 Answers: 0 Comments: 0

Question Number 194568 Answers: 1 Comments: 0

Equation.. J_𝛍 ^((1)) (z)Y_𝛍 (z)−J_𝛍 (z)Y_𝛍 ^((1)) (z)=−(2/(πz)) plz......Solve this Equation....... J_𝛍 (z) is First Kind Bessel Function Y_𝛍 (z) is Second Kind Bessel Function (aka Neuman Function) f^((n)) (z) n times derivate f(z) respect z

$$\mathrm{Equation}.. \\ $$$${J}_{\boldsymbol{\mu}} ^{\left(\mathrm{1}\right)} \left({z}\right){Y}_{\boldsymbol{\mu}} \left({z}\right)−{J}_{\boldsymbol{\mu}} \left({z}\right){Y}_{\boldsymbol{\mu}} ^{\left(\mathrm{1}\right)} \left({z}\right)=−\frac{\mathrm{2}}{\pi{z}} \\ $$$$\mathrm{plz}......\mathrm{Solve}\:\mathrm{this}\:\mathrm{Equation}....... \\ $$$${J}_{\boldsymbol{\mu}} \left({z}\right)\:\mathrm{is}\:\mathrm{First}\:\mathrm{Kind}\:\mathrm{Bessel}\:\mathrm{Function} \\ $$$${Y}_{\boldsymbol{\mu}} \left({z}\right)\:\mathrm{is}\:\mathrm{Second}\:\mathrm{Kind}\:\mathrm{Bessel}\:\mathrm{Function} \\ $$$$\left(\mathrm{aka}\:\mathrm{Neuman}\:\mathrm{Function}\right) \\ $$$${f}^{\left({n}\right)} \left({z}\right)\:{n}\:\mathrm{times}\:\mathrm{derivate}\:{f}\left({z}\right)\:\mathrm{respect}\:{z} \\ $$

Question Number 194564 Answers: 2 Comments: 0

soit A(2,1) B(3,2) C(4,3) points du plan( ox,oy) 1)Determiner l ′ equation du cercle qui passe par A; B; C ? 2) points d intersection du cercle avec l axe(ox,oy)?

$$\mathrm{soit}\:\:\:\boldsymbol{\mathrm{A}}\left(\mathrm{2},\mathrm{1}\right)\:\:\:\:\:\boldsymbol{\mathrm{B}}\left(\mathrm{3},\mathrm{2}\right)\:\:\:\:\:\boldsymbol{\mathrm{C}}\left(\mathrm{4},\mathrm{3}\right)\:\:\mathrm{points}\:\mathrm{du}\: \\ $$$$\:\:\mathrm{plan}\left(\:\mathrm{ox},\mathrm{oy}\right)\:\: \\ $$$$ \\ $$$$\left.\:\mathrm{1}\right)\mathrm{Determiner}\:\:\mathrm{l}\:'\:\mathrm{equation}\:\mathrm{du}\:\mathrm{cercle}\:\mathrm{qui}\: \\ $$$$\:\:\:\:\mathrm{passe}\:\mathrm{par}\:\boldsymbol{\mathrm{A}};\:\boldsymbol{\mathrm{B}};\:\boldsymbol{\mathrm{C}}\:\:\:? \\ $$$$\left.\:\:\mathrm{2}\right)\:\mathrm{points}\:\mathrm{d}\:\mathrm{intersection}\:\mathrm{du}\:\mathrm{cercle}\:\mathrm{avec} \\ $$$$\:\:\:\:\:\mathrm{l}\:\mathrm{axe}\left(\mathrm{ox},\mathrm{oy}\right)? \\ $$

Question Number 194563 Answers: 0 Comments: 0

A mass of 12kg rests on a smooth inclined plane which is 6m long and 1m high. The mass is connected by a light inextensible string which passes over a smooth pulley fixed at the top of the plane to a mass of 4kg which is hanging freely. With the string taut, the system is released from rest. Using Polya problem solving approach find the following: a. acceleration of the system b i. velocity with which the 4kg mass hits the ground. b ii. time the 4kg mass takes to hit the ground

Question Number 194560 Answers: 1 Comments: 0

Question Number 194559 Answers: 2 Comments: 0

repeat question Shiw that : Σ_(i=1) ^n ((1/(2i−1))−(1/(2i)))=Σ_(i=1) ^n (1/(n+i)) ?

$${repeat}\:{question} \\ $$$${Shiw}\:{that}\:: \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\frac{\mathrm{1}}{\mathrm{2}{i}−\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}{i}}\right)=\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{n}+{i}}\:\:? \\ $$

Question Number 194552 Answers: 2 Comments: 3

If a, b are real numbers & 4cos^2 θ = ((4a^2 +9b^2 +5)/(a+3b)), then the value of (a+b) will be: (a) (7/6) (b) (5/4) (c) ((11)/6) (d) ((17)/(12))

$$\mathrm{If}\:{a},\:{b}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\:\&\:\mathrm{4cos}^{\mathrm{2}} \theta\:=\:\frac{\mathrm{4}{a}^{\mathrm{2}} +\mathrm{9}{b}^{\mathrm{2}} +\mathrm{5}}{{a}+\mathrm{3}{b}},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left({a}+{b}\right)\:\mathrm{will}\:\mathrm{be}: \\ $$$$\left(\mathrm{a}\right)\:\frac{\mathrm{7}}{\mathrm{6}}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\frac{\mathrm{5}}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\frac{\mathrm{11}}{\mathrm{6}}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\frac{\mathrm{17}}{\mathrm{12}} \\ $$

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