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Question Number 87413 Answers: 1 Comments: 1

Question Number 87398 Answers: 0 Comments: 5

dear mr w a_(n+2) = a_(n+1) − a_n find a_n

$$\mathrm{dear}\:\mathrm{mr}\:\mathrm{w} \\ $$$$\mathrm{a}_{\mathrm{n}+\mathrm{2}} \:=\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:−\:\mathrm{a}_{\mathrm{n}} \\ $$$$\mathrm{find}\:\mathrm{a}_{\mathrm{n}} \\ $$

Question Number 87376 Answers: 1 Comments: 0

Question Number 87382 Answers: 1 Comments: 0

Show that tan((5π)/(12)) is a solution of this equation: x^3 −3x^2 −3x+1=0

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{tan}\frac{\mathrm{5}\pi}{\mathrm{12}}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{this}\: \\ $$$$\mathrm{equation}:\:{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 87379 Answers: 1 Comments: 2

Π_(k = 2) ^(2010) ((k^2 −1)/k^2 ) = ?

$$\underset{\mathrm{k}\:=\:\mathrm{2}} {\overset{\mathrm{2010}} {\prod}}\:\frac{\mathrm{k}^{\mathrm{2}} −\mathrm{1}}{\mathrm{k}^{\mathrm{2}} }\:=\:? \\ $$

Question Number 87378 Answers: 1 Comments: 1

lim_(x→0) ((tan 3x−3tan x)/x^3 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{3x}−\mathrm{3tan}\:\mathrm{x}}{\mathrm{x}^{\mathrm{3}} } \\ $$

Question Number 87371 Answers: 1 Comments: 3

∫((x^7 +x^3 +4)/(x^8 −x^5 +9))dx

$$\int\frac{{x}^{\mathrm{7}} +{x}^{\mathrm{3}} +\mathrm{4}}{{x}^{\mathrm{8}} −{x}^{\mathrm{5}} +\mathrm{9}}{dx} \\ $$

Question Number 87369 Answers: 1 Comments: 4

Question Number 87358 Answers: 0 Comments: 0

what is the number of integral values of y for which? lim_(x→0) ((y^2 +10y−24 sin ((1/x)))/(y^2 +10y −24 sin ((2/x)))) does not exist

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\: \\ $$$$\mathrm{integral}\:\mathrm{values}\:\mathrm{of}\:\mathrm{y}\:\mathrm{for}\:\mathrm{which}? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{y}^{\mathrm{2}} +\mathrm{10y}−\mathrm{24}\:\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{y}^{\mathrm{2}} +\mathrm{10y}\:−\mathrm{24}\:\mathrm{sin}\:\left(\frac{\mathrm{2}}{\mathrm{x}}\right)}\: \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$

Question Number 87352 Answers: 1 Comments: 6

f(x)= f(x+1) −x f(0)= 123 f(29) =?

$$\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)\:−\mathrm{x} \\ $$$$\mathrm{f}\left(\mathrm{0}\right)=\:\mathrm{123} \\ $$$$\mathrm{f}\left(\mathrm{29}\right)\:=? \\ $$

Question Number 87340 Answers: 3 Comments: 3

∫ ((cos x)/((5+4cos x)^2 )) dx =

$$\int\:\frac{\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{5}+\mathrm{4cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:= \\ $$

Question Number 87335 Answers: 1 Comments: 0

Question Number 87325 Answers: 1 Comments: 3

1) ∫e^(√x) dx 2)∫((√(sin(x)))/((√(sin(x)))+(√(cos(x)))))dx

$$\left.\mathrm{1}\right)\:\int{e}^{\sqrt{{x}}} \:{dx} \\ $$$$\left.\mathrm{2}\right)\int\frac{\sqrt{{sin}\left({x}\right)}}{\sqrt{{sin}\left({x}\right)}+\sqrt{{cos}\left({x}\right)}}{dx} \\ $$

Question Number 87322 Answers: 0 Comments: 0

Investigate the stationary value of (x^3 /(1+x^2 )) and sketch the graph

$$\mathrm{I}{nvestigate}\:{the}\:{stationary} \\ $$$${value}\:{of} \\ $$$$\frac{{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{and}\:{sketch}\:{the}\:{graph} \\ $$

Question Number 87321 Answers: 0 Comments: 0

Question Number 87316 Answers: 0 Comments: 0

Question Number 87313 Answers: 1 Comments: 4

a,b,c=1,2,3,...,n find Σ_(a≠b≠c) abc

$${a},{b},{c}=\mathrm{1},\mathrm{2},\mathrm{3},...,{n} \\ $$$${find}\:\underset{{a}\neq{b}\neq{c}} {\sum}{abc} \\ $$

Question Number 87308 Answers: 0 Comments: 0

A sequence (U_n ) is defined reculsively as U_o = (1/2) and U_(n+1) = (2/(1 + U_n )) for n ∈ N a) Show by mathematical induction that all terms in the sequence are positive. b) Given that the sequence (U_n ) is convergent, show that the limit,l, is a solution to the equation x^2 + x−2 = 0. Hence find l c) Given that (V_n ) is a sequence of general term such that V_n = ((U_n −1)/(U_n +2)) , ∀ n ∈ N. show that (V_n ) is convergent and determine its limit. hence deduce the convergence of the sequence (U_n ). Please recommend me textbooks for this topic even youtube vids please

$$\:\mathrm{A}\:\mathrm{sequence}\:\left({U}_{{n}} \right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{reculsively}\:\mathrm{as}\: \\ $$$$\:{U}_{{o}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{and}\:{U}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}}{\mathrm{1}\:+\:{U}_{{n}} }\:\mathrm{for}\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\left.\:\mathrm{a}\right)\:\mathrm{Show}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{all}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{sequence} \\ $$$$\:\:\:\:\:\mathrm{are}\:\mathrm{positive}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left({U}_{{n}} \right)\:\mathrm{is}\:\mathrm{convergent},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{limit},{l},\:\mathrm{is} \\ $$$$\:\:\:\:\mathrm{a}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{x}−\mathrm{2}\:=\:\mathrm{0}.\:\mathrm{Hence}\:\mathrm{find}\:{l} \\ $$$$\left.\:\mathrm{c}\right)\:\:\mathrm{Given}\:\mathrm{that}\:\left({V}_{{n}} \right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{general}\:\mathrm{term}\:\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:{V}_{{n}} \:=\:\frac{{U}_{{n}} −\mathrm{1}}{{U}_{{n}} +\mathrm{2}}\:,\:\forall\:{n}\:\in\:\mathbb{N}. \\ $$$$\:\:\mathrm{show}\:\mathrm{that}\:\left({V}_{{n}} \right)\:\mathrm{is}\:\mathrm{convergent}\:\mathrm{and}\:\mathrm{determine}\:\mathrm{its}\:\:\mathrm{limit}. \\ $$$$\mathrm{hence}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\left({U}_{{n}} \right). \\ $$$$\:\:{Please}\:{recommend}\:{me}\:{textbooks}\:{for}\:{this}\:{topic}\:{even}\:{youtube}\:{vids} \\ $$$${please} \\ $$$$ \\ $$

Question Number 87306 Answers: 1 Comments: 9

calculate by complex method ∫_1 ^(+∞) ((xdx)/(x^4 +1))

$${calculate}\:{by}\:{complex}\:{method}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{xdx}}{{x}^{\mathrm{4}} \:+\mathrm{1}} \\ $$

Question Number 87302 Answers: 1 Comments: 3

for ∣z−1∣=1 show that tan(((arg(z−1))/2))−((2i)/z)=−1

$${for}\:\mid{z}−\mathrm{1}\mid=\mathrm{1}\:{show}\:{that} \\ $$$${tan}\left(\frac{{arg}\left({z}−\mathrm{1}\right)}{\mathrm{2}}\right)−\frac{\mathrm{2}{i}}{{z}}=−\mathrm{1} \\ $$

Question Number 87301 Answers: 0 Comments: 3