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Question Number 194709 Answers: 1 Comments: 0

Show that in fibonacci sequence f_(3n) =f_n ^3 +f_(n+1) ^3 −f_(n−1) ^3

$${Show}\:{that}\:\:{in}\:{fibonacci}\:{sequence} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{f}_{\mathrm{3}{n}} ={f}_{{n}} ^{\mathrm{3}} +{f}_{{n}+\mathrm{1}} ^{\mathrm{3}} −{f}_{{n}−\mathrm{1}} ^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 194710 Answers: 0 Comments: 21

let p be a prime number & let a_1 ,a_2 ,a_3 ,...,a_(p ) be integers show that , there exists an integer k such that the numbers a_1 +k, a_2 +k,a_3 +k,....,a_p +k produce at least (1/2)p distinct remainders when divided by p.

$${let}\:{p}\:{be}\:{a}\:{prime}\:{number} \\ $$$$\&\:{let}\:{a}_{\mathrm{1}} \:,{a}_{\mathrm{2}} ,{a}_{\mathrm{3}} ,...,{a}_{{p}\:} {be}\:{integers} \\ $$$${show}\:{that}\:,\:{there}\:{exists}\:{an}\:{integer}\:{k}\:{such}\:{that}\:{the}\:{numbers} \\ $$$${a}_{\mathrm{1}} +{k},\:{a}_{\mathrm{2}} +{k},{a}_{\mathrm{3}} +{k},....,{a}_{{p}} +{k} \\ $$$${produce}\:{at}\:{least}\:\frac{\mathrm{1}}{\mathrm{2}}{p}\:{distinct}\:{remainders} \\ $$$${when}\:{divided}\:{by}\:{p}. \\ $$

Question Number 194700 Answers: 0 Comments: 2

Question Number 194697 Answers: 2 Comments: 0

((tan x)/(tan x−tan 3x)) = (1/3) then ((cot x)/(cot x+cot 3x)) =?

$$\:\:\: \frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{tan}\:\mathrm{x}−\mathrm{tan}\:\mathrm{3x}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{then} \\ $$$$\:\:\:\frac{\mathrm{cot}\:\mathrm{x}}{\mathrm{cot}\:\mathrm{x}+\mathrm{cot}\:\mathrm{3x}}\:=? \\ $$

Question Number 194695 Answers: 1 Comments: 0

(x/(a+b−c)) =(y/(b+c−a))=(z/(c+a−b)) Then (a−b)x+(b−c)y+(c−a)z =?

$$\:\:\:\: \:\frac{\mathrm{x}}{\mathrm{a}+\mathrm{b}−\mathrm{c}}\:=\frac{\mathrm{y}}{\mathrm{b}+\mathrm{c}−\mathrm{a}}=\frac{\mathrm{z}}{\mathrm{c}+\mathrm{a}−\mathrm{b}} \\ $$$$\:\mathrm{Then}\:\left(\mathrm{a}−\mathrm{b}\right)\mathrm{x}+\left(\mathrm{b}−\mathrm{c}\right)\mathrm{y}+\left(\mathrm{c}−\mathrm{a}\right)\mathrm{z}\:=? \\ $$

Question Number 194693 Answers: 1 Comments: 0

if f_n =f_(n−1) +f_(n−2) ; f_1 =f_2 =1 then prove that 5∣f_(5n)

$${if}\:\:\:{f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \:\:;\:\:{f}_{\mathrm{1}} ={f}_{\mathrm{2}} =\mathrm{1} \\ $$$${then}\:\:\:{prove}\:{that}\:\:\:\mathrm{5}\mid{f}_{\mathrm{5}{n}} \:\: \\ $$

Question Number 194685 Answers: 1 Comments: 0

Question Number 194662 Answers: 0 Comments: 2

∫_0 ^(Π/2) (√(4sin^2 t+cos^2 t)) dt

$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}}\:\:{dt} \\ $$

Question Number 194654 Answers: 1 Comments: 0

Question Number 194652 Answers: 0 Comments: 0

Question Number 194649 Answers: 0 Comments: 3

calcul ∫_0 ^(Π/2) (√(4sin^2 t+cos^2 t ))dt

$${calcul}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}\:}{dt} \\ $$

Question Number 194648 Answers: 3 Comments: 3

Question Number 194642 Answers: 2 Comments: 0

If A= (((a b c)),((b c a)),((c a b)) ) and a,b,c >0 such that abc=1 and A^T .A=I find a^3 +b^3 +c^3 −3abc .

$$\:\mathrm{If}\:\mathrm{A}=\begin{pmatrix}{\mathrm{a}\:\:\:\:\mathrm{b}\:\:\:\:\:\:\mathrm{c}}\\{\mathrm{b}\:\:\:\:\mathrm{c}\:\:\:\:\:\:\mathrm{a}}\\{\mathrm{c}\:\:\:\:\:\mathrm{a}\:\:\:\:\:\:\mathrm{b}}\end{pmatrix}\:\mathrm{and}\:\mathrm{a},\mathrm{b},\mathrm{c}\:>\mathrm{0} \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{abc}=\mathrm{1}\:\mathrm{and}\:\mathrm{A}^{\mathrm{T}} .\mathrm{A}=\mathrm{I} \\ $$$$\:\mathrm{find}\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} −\mathrm{3abc}\:. \\ $$

Question Number 194640 Answers: 1 Comments: 0

lim_(x→∞) ((√x) (1−cos ((1/(2(√x))))).(1/(sin ((3/( (√x)))))))

$$\:\:\: \: \\ $$$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt{\mathrm{x}}\:\left(\mathrm{1}−\mathrm{cos}\:\left(\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{x}}}\right)\right).\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\mathrm{3}}{\:\sqrt{\mathrm{x}}}\right)}\right)\: \\ $$

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.

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