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Question Number 70132    Answers: 1   Comments: 1

Σ_(n=1) ^(3050) i^n

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{3050}} {\sum}}\:{i}^{{n}} \\ $$

Question Number 70069    Answers: 1   Comments: 2

Π_(n=1) ^5 (((12n−2)^4 +18^2 )/((12n−8)^4 +18^2 )) =(((10^4 +324)(22^4 +324)(34^4 +324)(46^4 +324)(58^4 +324))/((4^4 +324)(16^4 +324)(28^4 +324)(40^4 +324)(52^4 +324)))

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\prod}}\frac{\left(\mathrm{12}{n}−\mathrm{2}\right)^{\mathrm{4}} +\mathrm{18}^{\mathrm{2}} }{\left(\mathrm{12}{n}−\mathrm{8}\right)^{\mathrm{4}} +\mathrm{18}^{\mathrm{2}} } \\ $$$$=\frac{\left(\mathrm{10}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{22}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{34}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{46}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{58}^{\mathrm{4}} +\mathrm{324}\right)}{\left(\mathrm{4}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{16}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{28}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{40}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{52}^{\mathrm{4}} +\mathrm{324}\right)} \\ $$

Question Number 70035    Answers: 0   Comments: 4

Question Number 70030    Answers: 1   Comments: 0

Find the convergence of Σ_(n=1) ^∞ (((1/n) + 1)/(−n^2 ))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{of} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\frac{\mathrm{1}}{{n}}\:+\:\mathrm{1}}{−{n}^{\mathrm{2}} } \\ $$

Question Number 70051    Answers: 2   Comments: 0

((a+b)/c)=((cos(((a−b)/2)))/(cos(c/2)))

$$\frac{{a}+{b}}{{c}}=\frac{{cos}\left(\frac{{a}−{b}}{\mathrm{2}}\right)}{{cos}\frac{{c}}{\mathrm{2}}} \\ $$

Question Number 70022    Answers: 0   Comments: 1

Find all pairs of (p, q) integer(s) such that p^3 − q^5 = (p + q)^2

$${Find}\:\:\:{all}\:\:{pairs}\:\:{of}\:\:\:\left({p},\:{q}\right)\:\:{integer}\left({s}\right)\:\:{such}\:\:{that} \\ $$$${p}^{\mathrm{3}} \:−\:{q}^{\mathrm{5}} \:\:=\:\:\left({p}\:+\:{q}\right)^{\mathrm{2}} \\ $$

Question Number 70017    Answers: 0   Comments: 2

Solution- log_8 x+log_4 x+log_2 x=11 ⇒(1/(log_x 8))+(1/(log_x 4))+(1/(log_x 2))=11 ⇒(1/(log_x 2^3 ))+(1/(log_x 2^2 ))+(1/(log_x 2))=11 ⇒(1/(3log_x 2))+(1/(2log_x 2))+(1/(log_x 2))=11 ⇒((1/3)+(1/2)+1)(1/(log_x 2))=11 ⇒((11)/6)×(1/(log_x 2))=11 ⇒(1/(log_x 2))=11×(6/(11)) ⇒log_2 x=6 ⇒x=2^6 ∴x=64 is this rule correct????

$$\mathrm{Solution}- \\ $$$$\mathrm{log}_{\mathrm{8}} \mathrm{x}+\mathrm{log}_{\mathrm{4}} \mathrm{x}+\mathrm{log}_{\mathrm{2}} \mathrm{x}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{8}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{4}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{3log}_{\mathrm{x}} \mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2log}_{\mathrm{x}} \mathrm{2}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow\left(\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}\right)\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{11}}{\mathrm{6}}×\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11}×\frac{\mathrm{6}}{\mathrm{11}} \\ $$$$\Rightarrow\mathrm{log}_{\mathrm{2}} \mathrm{x}=\mathrm{6} \\ $$$$\Rightarrow\mathrm{x}=\mathrm{2}^{\mathrm{6}} \\ $$$$\therefore\mathrm{x}=\mathrm{64} \\ $$$$ \\ $$$$\mathrm{is}\:\mathrm{this}\:\mathrm{rule}\:\mathrm{correct}???? \\ $$

Question Number 69894    Answers: 0   Comments: 3

∫ ((2x^5 −x)/(x^3 −2))dx

$$\int\:\frac{\mathrm{2}{x}^{\mathrm{5}} −{x}}{{x}^{\mathrm{3}} −\mathrm{2}}{dx} \\ $$

Question Number 69829    Answers: 2   Comments: 2

Question Number 69778    Answers: 2   Comments: 5

prove that the equation (b^2 −4ac)x^2 + 4(a + c)x −4 = 0 is always real.

$${prove}\:{that}\:{the}\:{equation}\: \\ $$$$\:\:\left({b}^{\mathrm{2}} −\mathrm{4}{ac}\right){x}^{\mathrm{2}} \:+\:\mathrm{4}\left({a}\:+\:{c}\right){x}\:−\mathrm{4}\:=\:\mathrm{0}\:{is}\:{always}\:{real}. \\ $$

Question Number 69766    Answers: 0   Comments: 4

find (dy/dx) at the point (0,3) when 2x^2 y + y + 4xy^2 = 2x + 3

$${find}\:\:\frac{{dy}}{{dx}}\:\:{at}\:{the}\:{point}\:\:\left(\mathrm{0},\mathrm{3}\right)\:\:{when}\:\:\mathrm{2}{x}^{\mathrm{2}} {y}\:+\:{y}\:+\:\mathrm{4}{xy}^{\mathrm{2}} \:=\:\mathrm{2}{x}\:+\:\mathrm{3}\: \\ $$

Question Number 69765    Answers: 1   Comments: 0

Given that y = (√(5x^2 + 3)) , show that when x^2 = (6/5) , (d^2 y/dx^(2 ) ) = ((125)/8)

$${Given}\:{that}\:\:{y}\:=\:\sqrt{\mathrm{5}{x}^{\mathrm{2}} \:+\:\mathrm{3}}\:,\:{show}\:{that}\:\:{when}\:{x}^{\mathrm{2}} \:=\:\frac{\mathrm{6}}{\mathrm{5}}\:,\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}\:} }\:=\:\frac{\mathrm{125}}{\mathrm{8}} \\ $$

Question Number 69764    Answers: 1   Comments: 1

find (dy/dx) if x = sin^2 t and y= tan t at t = (π/4)

$${find}\:\:\:\frac{{dy}}{{dx}}\:\:{if}\:\:{x}\:=\:{sin}^{\mathrm{2}} {t}\:\:{and}\:\:{y}=\:{tan}\:{t}\:{at}\:\:{t}\:=\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 69763    Answers: 1   Comments: 2

find (dy/dx) if y = 3^x e^(2x + 1) , at x =1

$${find}\:\frac{{dy}}{{dx}}\:\:{if}\:\:{y}\:=\:\mathrm{3}^{{x}} {e}^{\mathrm{2}{x}\:+\:\mathrm{1}} ,\:{at}\:{x}\:=\mathrm{1} \\ $$

Question Number 69762    Answers: 1   Comments: 1

prove by mathematical induction, that for all positive integers n, Σ_(r=1) ^n r(r + 1) = (n/3)(n + 1)( n + 2)

$${prove}\:{by}\:{mathematical}\:{induction},\:{that}\:{for}\:{all}\:{positive}\:{integers}\:{n}, \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left(\:{n}\:+\:\mathrm{2}\right) \\ $$

Question Number 69715    Answers: 1   Comments: 0

∫((x^2 +1)/((x+1)^2 ))e^x dx

$$\int\frac{{x}^{\mathrm{2}} +\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{e}^{{x}} {dx} \\ $$

Question Number 69576    Answers: 1   Comments: 1

∫_(−2) ^( 2) (x^3 cos(x/2)+(1/2))(√(4−x^2 ))dx

$$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \left({x}^{\mathrm{3}} {cos}\frac{{x}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 69459    Answers: 0   Comments: 0

Question Number 69314    Answers: 1   Comments: 1

if 5 x y 40 are in GP .find x and y

$${if}\:\mathrm{5}\:{x}\:{y}\:\mathrm{40}\:{are}\:{in}\:{GP}\:.{find}\:{x}\:{and}\:{y} \\ $$

Question Number 69247    Answers: 0   Comments: 1

show that c∣a ⇔ −c∣a.

$${show}\:{that}\: \\ $$$$\:{c}\mid{a}\:\Leftrightarrow\:−{c}\mid{a}. \\ $$

Question Number 69236    Answers: 0   Comments: 0

Use Residus theorem to prove that ∀ a>0 Σ_(n=0) ^∞ (1/( n^2 +a^2 )) = (1/2)((π/(ash(πa))) −(1/a^2 )) and Σ_(n=0) ^∞ (((−1)^n )/(n^2 +a^2 )) = (1/2)((( π)/(a.th(πa))) −(1/a^2 )) Assume that we can developp in integer serie the functions f(x)=(x/(shx)) and g(x)=(x/(thx)) Give the DL_2 of f and g around zero Why can′t we use that theorem to explicit f(a)=Σ_(n=0) ^∞ (((−1)^n )/( (2n+1)^2 +a^2 )) ???

$${Use}\:\:{Residus}\:{theorem}\:{to}\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\pi}{{ash}\left(\pi{a}\right)}\:\:\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$${and}\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\:\pi}{{a}.{th}\left(\pi{a}\right)}\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$$\:\:\:{Assume}\:{that}\:{we}\:{can}\:{developp}\:{in}\:{integer}\:{serie}\:{the}\:{functions} \\ $$$${f}\left({x}\right)=\frac{{x}}{{shx}}\:\:\:{and}\:{g}\left({x}\right)=\frac{{x}}{{thx}}\: \\ $$$$\:{Give}\:{the}\:{DL}_{\mathrm{2}} \:{of}\:\:{f}\:{and}\:{g}\:{around}\:{zero}\: \\ $$$${Why}\:{can}'{t}\:{we}\:{use}\:{that}\:{theorem}\:{to}\:{explicit} \\ $$$${f}\left({a}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\:\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} +{a}^{\mathrm{2}} }\:\:\:??? \\ $$

Question Number 69207    Answers: 1   Comments: 3

help please. A river is 5m wide and flows at 3.0ms^(−1) . A man can swim at 2.0ms^(−1) in still water. if he sets off at an angle of 90° to the bank calculate a) the mans time and velocity b) his distance downstream from the starting point till when he reaches the other side of the river bank c) the actual distance he swims through the water.

$${help}\:{please}. \\ $$$$ \\ $$$${A}\:{river}\:{is}\:\mathrm{5}{m}\:{wide}\:{and}\:{flows}\:{at}\:\mathrm{3}.\mathrm{0}{ms}^{−\mathrm{1}} .\:{A}\:{man}\:{can}\:{swim}\:{at}\:\mathrm{2}.\mathrm{0}{ms}^{−\mathrm{1}} \\ $$$${in}\:{still}\:{water}.\:{if}\:{he}\:{sets}\:{off}\:{at}\:{an}\:{angle}\:{of}\:\mathrm{90}°\:{to}\:{the}\:{bank} \\ $$$${calculate} \\ $$$$\left.{a}\right)\:{the}\:{mans}\:{time}\:{and}\:{velocity} \\ $$$$\left.{b}\right)\:{his}\:{distance}\:{downstream}\:{from}\:{the}\:{starting}\:{point}\:{till} \\ $$$${when}\:{he}\:{reaches}\:{the}\:{other}\:{side}\:{of}\:{the}\:{river}\:{bank} \\ $$$$\left.{c}\right)\:{the}\:{actual}\:{distance}\:{he}\:{swims}\:{through}\:{the}\:{water}. \\ $$

Question Number 69198    Answers: 0   Comments: 0

please someone check Q69159

$$\:{please}\:{someone}\:{check}\:\:{Q}\mathrm{69159} \\ $$

Question Number 69188    Answers: 2   Comments: 2

Question Number 69092    Answers: 1   Comments: 0

Question Number 69106    Answers: 0   Comments: 0

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