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Question Number 57060    Answers: 0   Comments: 0

Question Number 57051    Answers: 0   Comments: 5

F_n = ((ϕ^n − (1 − ϕ)^n )/(√5)) , with { ((ϕ = ((1 + (√5))/2))) :} (G_n ) : { ((G_1 = 1)),((G_2 = 1)),((G_m = G_(m−1) + G_(m−2) )) :} Give a proof for F_n = G_n , ∀n∈N^∗ . Thank you

$$ \\ $$$$\:\:\boldsymbol{{F}}_{\boldsymbol{{n}}} \:=\:\frac{\varphi^{{n}} \:−\:\left(\mathrm{1}\:−\:\varphi\right)^{{n}} }{\sqrt{\mathrm{5}}}\:,\:\:\mathrm{with}\:\begin{cases}{\varphi\:=\:\frac{\mathrm{1}\:+\:\sqrt{\mathrm{5}}}{\mathrm{2}}}\end{cases} \\ $$$$ \\ $$$$\:\:\left(\boldsymbol{{G}}_{\boldsymbol{{n}}} \right)\::\:\begin{cases}{\boldsymbol{{G}}_{\mathrm{1}} \:=\:\mathrm{1}}\\{\boldsymbol{{G}}_{\mathrm{2}} \:=\:\mathrm{1}}\\{\boldsymbol{{G}}_{{m}} \:=\:\boldsymbol{{G}}_{\mathrm{m}−\mathrm{1}} \:+\:\boldsymbol{{G}}_{{m}−\mathrm{2}} }\end{cases} \\ $$$$ \\ $$$$\:\:\mathrm{Give}\:\mathrm{a}\:\mathrm{proof}\:\mathrm{for}\:\boldsymbol{{F}}_{\boldsymbol{{n}}} \:=\:\boldsymbol{{G}}_{\boldsymbol{{n}}} \:,\:\forall{n}\in\mathbb{N}^{\ast} .\: \\ $$$$\:\:\mathrm{Thank}\:\mathrm{you} \\ $$

Question Number 57048    Answers: 2   Comments: 1

Question Number 57046    Answers: 0   Comments: 1

Question Number 57044    Answers: 1   Comments: 1

Question Number 57042    Answers: 1   Comments: 1

Question Number 57024    Answers: 0   Comments: 1

If f(x−1)=2x^3 −3x^2 +7x+10. Find f(3).

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}−\mathrm{1}\right)=\mathrm{2x}^{\mathrm{3}} −\mathrm{3x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{10}.\:\mathrm{Find}\:\mathrm{f}\left(\mathrm{3}\right). \\ $$

Question Number 57023    Answers: 1   Comments: 0

There are 28 players in a national football team. 14 play midfield and defence, 15 play defence and attack and 3 play midfield only. the number of players who play attack only is twice those who play defence only, and the number who play defence is equal to those who play attack. If 18 play midfield represent the information on a venn diagram. Find i. how many play at most two games ii. how many play neither attack nor defence iii. how many play either midfield or attack.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{28}\:\mathrm{players}\:\mathrm{in}\:\mathrm{a}\:\mathrm{national}\: \\ $$$$\mathrm{football}\:\mathrm{team}.\:\mathrm{14}\:\mathrm{play}\:\mathrm{midfield}\:\mathrm{and} \\ $$$$\mathrm{defence},\:\mathrm{15}\:\mathrm{play}\:\mathrm{defence}\:\mathrm{and}\:\mathrm{attack}\: \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{play}\:\mathrm{midfield}\:\mathrm{only}.\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{players}\:\mathrm{who}\:\mathrm{play}\:\mathrm{attack}\:\mathrm{only}\:\mathrm{is}\: \\ $$$$\mathrm{twice}\:\mathrm{those}\:\mathrm{who}\:\mathrm{play}\:\mathrm{defence}\:\mathrm{only},\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{who}\:\mathrm{play}\:\mathrm{defence} \\ $$$$\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{those}\:\mathrm{who}\:\mathrm{play}\:\mathrm{attack}.\:\mathrm{If}\: \\ $$$$\mathrm{18}\:\mathrm{play}\:\mathrm{midfield}\:\mathrm{represent}\:\mathrm{the}\: \\ $$$$\mathrm{information}\:\mathrm{on}\:\mathrm{a}\:\mathrm{venn}\:\mathrm{diagram}. \\ $$$$\mathrm{Find} \\ $$$$\mathrm{i}.\:\mathrm{how}\:\mathrm{many}\:\mathrm{play}\:\mathrm{at}\:\mathrm{most}\:\mathrm{two}\:\mathrm{games} \\ $$$$\mathrm{ii}.\:\mathrm{how}\:\mathrm{many}\:\mathrm{play}\:\mathrm{neither}\:\mathrm{attack}\:\mathrm{nor} \\ $$$$\mathrm{defence} \\ $$$$\mathrm{iii}.\:\mathrm{how}\:\mathrm{many}\:\mathrm{play}\:\mathrm{either}\:\mathrm{midfield}\:\mathrm{or}\: \\ $$$$\mathrm{attack}. \\ $$

Question Number 57020    Answers: 0   Comments: 3

[f(x+1)−f(x)]^2 =4[f(x)−1] f(x)=? −−−−−−−−−−−−− f(0)=0

$$\left[{f}\left({x}+\mathrm{1}\right)−{f}\left({x}\right)\right]^{\mathrm{2}} =\mathrm{4}\left[{f}\left({x}\right)−\mathrm{1}\right] \\ $$$${f}\left({x}\right)=? \\ $$$$−−−−−−−−−−−−− \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$

Question Number 57012    Answers: 2   Comments: 0

Find the local minimum value of f(x) where f(x)= (x−(1/x))+((2/(x−(1/x)))) ?

$${Find}\:{the}\:{local}\:{minimum}\:{value}\:{of}\:{f}\left({x}\right) \\ $$$${where}\:{f}\left({x}\right)=\:\left({x}−\frac{\mathrm{1}}{{x}}\right)+\left(\frac{\mathrm{2}}{{x}−\frac{\mathrm{1}}{{x}}}\right)\:? \\ $$

Question Number 57011    Answers: 1   Comments: 0

f(((x+y)/2))f(((x−y)/2))=g(x) g(x+y)g(x−y)=[f(x)]^2 −[f(y)]^2 f(x),g(x)=?

$${f}\left(\frac{{x}+{y}}{\mathrm{2}}\right){f}\left(\frac{{x}−{y}}{\mathrm{2}}\right)={g}\left({x}\right) \\ $$$${g}\left({x}+{y}\right){g}\left({x}−{y}\right)=\left[{f}\left({x}\right)\right]^{\mathrm{2}} −\left[{f}\left({y}\right)\right]^{\mathrm{2}} \\ $$$${f}\left({x}\right),{g}\left({x}\right)=? \\ $$

Question Number 57010    Answers: 1   Comments: 1

f(((x+y)/2))=((f(x)f(y))/(f(2))) f(x)=?

$${f}\left(\frac{{x}+{y}}{\mathrm{2}}\right)=\frac{{f}\left({x}\right){f}\left({y}\right)}{{f}\left(\mathrm{2}\right)} \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 57007    Answers: 1   Comments: 3

cosec ((π/(14))) − 4 cos (((2π)/7)) = ?

$$\mathrm{cosec}\:\left(\frac{\pi}{\mathrm{14}}\right)\:−\:\mathrm{4}\:\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\:\:=\:\:? \\ $$

Question Number 57001    Answers: 0   Comments: 1

construct an analytic function f(z) whose real part is e^x cos y

$${construct}\:{an}\:{analytic}\:{function}\:{f}\left({z}\right)\:{whose}\:{real}\:{part}\:{is}\:{e}^{{x}} \mathrm{cos}\:{y} \\ $$

Question Number 57000    Answers: 3   Comments: 1

If (x+2)^2 is a factor of the polynomial f(x)=mx^3 +x^2 +x+n, find; the values of m and n.

$$\mathrm{If}\:\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{2}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{mx}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{n},\:\mathrm{find}; \\ $$$$\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}. \\ $$

Question Number 56991    Answers: 0   Comments: 4

x! − x^2 = 8 , Find x

$$\:\:\mathrm{x}!\:−\:\mathrm{x}^{\mathrm{2}} \:\:=\:\:\mathrm{8}\:,\:\:\:\:\mathrm{Find}\:\:\mathrm{x} \\ $$$$ \\ $$

Question Number 56986    Answers: 2   Comments: 1

Question Number 56971    Answers: 1   Comments: 1

Question Number 56962    Answers: 1   Comments: 1

find S_n =Σ_(k=0) ^n k^2 C_n ^k cos(2kx) interms of n.

$${find}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \:{cos}\left(\mathrm{2}{kx}\right) \\ $$$${interms}\:{of}\:{n}. \\ $$

Question Number 56961    Answers: 1   Comments: 0

Out of 6 mathematicians and 7 physicists a committee consisting of 3 mathematicians and 3 physicists is to be formed. In how many ways can this be done if two particular mathematicians cannot be on the commitee?

$$\mathrm{Out}\:\mathrm{of}\:\mathrm{6}\:\mathrm{mathematicians}\:\mathrm{and}\:\mathrm{7}\:\mathrm{physicists} \\ $$$$\mathrm{a}\:\mathrm{committee}\:\mathrm{consisting}\:\mathrm{of}\:\mathrm{3}\:\mathrm{mathematicians} \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{physicists}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}.\:\mathrm{In}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{done}\:\mathrm{if}\:\mathrm{two}\: \\ $$$$\mathrm{particular}\:\mathrm{mathematicians}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{commitee}? \\ $$

Question Number 56954    Answers: 2   Comments: 1

Find minimum value of : cos(ω−φ)+cos(φ−ϕ)+cos (ϕ−ω).

$${Find}\:{minimum}\:{value}\:{of}\:: \\ $$$${cos}\left(\omega−\phi\right)+\mathrm{cos}\left(\phi−\varphi\right)+\mathrm{cos}\:\left(\varphi−\omega\right). \\ $$

Question Number 56951    Answers: 0   Comments: 2

The deviations from the mean of a set of numbers are (x+2), (2x−11), −9, (x+1)^2 , (x−4)^2 , (1−3x). find the value of x where x>0.

$$\mathrm{The}\:\mathrm{deviations}\:\mathrm{from}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{set}\:\mathrm{of}\:\mathrm{numbers}\:\mathrm{are}\:\left(\mathrm{x}+\mathrm{2}\right),\:\left(\mathrm{2x}−\mathrm{11}\right), \\ $$$$−\mathrm{9},\:\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} ,\:\left(\mathrm{x}−\mathrm{4}\right)^{\mathrm{2}} ,\:\left(\mathrm{1}−\mathrm{3x}\right).\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{where}\:\mathrm{x}>\mathrm{0}. \\ $$

Question Number 56949    Answers: 0   Comments: 1

Question Number 56939    Answers: 1   Comments: 2

calculate ∫ (dx/((x+1)^3 (x^2 −3x +2))) 2) find the value of ∫_2 ^(+∞) (dx/((x+1)^3 (x^2 −3x+2)))

$${calculate}\:\int\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{3}{x}\:+\mathrm{2}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{2}} ^{+\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}\right)} \\ $$

Question Number 56938    Answers: 0   Comments: 0

let A_n =∫∫_W_n e^(−xy) (√(x^2 +y^2 ))dxdy with W_n =[(1/n),n[×[(1/n),n[ 1) find A_n interms of n 2) determine lim_(n→+∞) A_n

$${let}\:{A}_{{n}} =\int\int_{{W}_{{n}} } {e}^{−{xy}} \sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy}\:\:\:{with}\:{W}_{{n}} =\left[\frac{\mathrm{1}}{{n}},{n}\left[×\left[\frac{\mathrm{1}}{{n}},{n}\left[\right.\right.\right.\right. \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$$ \\ $$

Question Number 56937    Answers: 0   Comments: 0

1. calculate f(x) =∫_0 ^(π/4) ln(1+xtanθ)dθ 2. calculate ∫_0 ^1 f(x)dx

$$\mathrm{1}.\:{calculate}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{ln}\left(\mathrm{1}+{xtan}\theta\right){d}\theta \\ $$$$\mathrm{2}.\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$

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