Question and Answers Forum

AllQuestion and Answers: Page 748

Question Number 143393 Answers: 1 Comments: 0

Given that f(r)=(r+1)! ∙ r, show that f(r)−f(r−1)=r!(r^2 +1). Hence or otherwise, show that 2! ∙ 5+3! ∙ 10+4! ∙ 17+......n!(n^2 +1)=(n+1)! ∙ (n−2)

$$\mathrm{Given}\:\mathrm{that}\:{f}\left({r}\right)=\left({r}+\mathrm{1}\right)!\:\centerdot\:{r},\:\mathrm{show}\:\mathrm{that} \\ $$$${f}\left({r}\right)−{f}\left({r}−\mathrm{1}\right)={r}!\left({r}^{\mathrm{2}} +\mathrm{1}\right). \\ $$$$\mathrm{Hence}\:\mathrm{or}\:\mathrm{otherwise},\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{2}!\:\centerdot\:\mathrm{5}+\mathrm{3}!\:\centerdot\:\mathrm{10}+\mathrm{4}!\:\centerdot\:\mathrm{17}+......{n}!\left({n}^{\mathrm{2}} +\mathrm{1}\right)=\left({n}+\mathrm{1}\right)!\:\centerdot\:\left({n}−\mathrm{2}\right) \\ $$

Question Number 143391 Answers: 1 Comments: 0

lim_(n→∞) (1 − (n/(n−2)))^(4n) = ?? chiaha daniel

$${lim}_{{n}\rightarrow\infty} \left(\mathrm{1}\:−\:\frac{{n}}{{n}−\mathrm{2}}\right)^{\mathrm{4}{n}} =\:?? \\ $$$${chiaha}\:{daniel} \\ $$

Question Number 143390 Answers: 1 Comments: 0

Given that (((tan α)/(sin θ))−((tan β)/(tan θ)))^2 =tan^2 α−tan^2 β, prove that cos θ=((tan β)/(tan α ))

$$\mathrm{Given}\:\mathrm{that}\:\left(\frac{\mathrm{tan}\:\alpha}{\mathrm{sin}\:\theta}−\frac{\mathrm{tan}\:\beta}{\mathrm{tan}\:\theta}\right)^{\mathrm{2}} =\mathrm{tan}^{\mathrm{2}} \alpha−\mathrm{tan}^{\mathrm{2}} \beta, \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{cos}\:\theta=\frac{\mathrm{tan}\:\beta}{\mathrm{tan}\:\alpha\:} \\ $$

Question Number 143384 Answers: 2 Comments: 0

proof: tg^2 (36°) ∙ tg^2 (72°) = 5

$${proof}:\:\:{tg}^{\mathrm{2}} \left(\mathrm{36}°\right)\:\centerdot\:{tg}^{\mathrm{2}} \left(\mathrm{72}°\right)\:=\:\mathrm{5} \\ $$

Question Number 143383 Answers: 0 Comments: 0

calculate ∫_1 ^3 ((√x)/( (√(x+1))+(√(2x+3))))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\frac{\sqrt{{x}}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{\mathrm{2}{x}+\mathrm{3}}}{dx} \\ $$

Question Number 143382 Answers: 0 Comments: 0

find ∫_0 ^∞ xe^(−x^2 ) log(1+2x^2 )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} {xe}^{−{x}^{\mathrm{2}} } {log}\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 143381 Answers: 2 Comments: 0

let f(x)=arctan((√2)x^2 ) 1) calculate f^((n)) (x)and f^((n)) (0) 2)if f(x)=Σa_n x^n find the sequence a_n

$${let}\:{f}\left({x}\right)={arctan}\left(\sqrt{\mathrm{2}}{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right){and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){if}\:{f}\left({x}\right)=\Sigma{a}_{{n}} {x}^{{n}} \:\:{find}\:{the}\: \\ $$$${sequence}\:{a}_{{n}} \\ $$

Question Number 143380 Answers: 1 Comments: 0

developp at fourier serie f(x)=(3/(1+2cosx)) by use of two methods

$${developp}\:{at}\:{fourier}\:{serie} \\ $$$${f}\left({x}\right)=\frac{\mathrm{3}}{\mathrm{1}+\mathrm{2}{cosx}} \\ $$$${by}\:{use}\:{of}\:{two}\:{methods} \\ $$

Question Number 143372 Answers: 1 Comments: 1

Question Number 143370 Answers: 0 Comments: 0

(32)^((log_8 X^3 )^3 ) +(80)^((log_4 X^2 )^2 ) −(144)^((log_2 X)) =18 find X

$$\left(\mathrm{32}\right)^{\left({log}_{\mathrm{8}} {X}^{\mathrm{3}} \right)^{\mathrm{3}} } +\left(\mathrm{80}\right)^{\left({log}_{\mathrm{4}} {X}^{\mathrm{2}} \right)^{\mathrm{2}} } −\left(\mathrm{144}\right)^{\left({log}_{\mathrm{2}} {X}\right)} =\mathrm{18} \\ $$$${find}\:{X} \\ $$

Question Number 143369 Answers: 0 Comments: 0

Question Number 143368 Answers: 2 Comments: 1

Question Number 143367 Answers: 2 Comments: 0

Question Number 143350 Answers: 1 Comments: 0

if tan^2 αtan^2 β+tan^2 βtan^2 γ+ tan^2 γtan^2 α+2tan^2 αtan^2 βtan^2 γ=1 then find sin^2 α+sin^2 β+sin^2 γ

$$\mathrm{if}\:\mathrm{tan}^{\mathrm{2}} \alpha\mathrm{tan}^{\mathrm{2}} \beta+\mathrm{tan}^{\mathrm{2}} \beta\mathrm{tan}^{\mathrm{2}} \gamma+ \\ $$$$\mathrm{tan}^{\mathrm{2}} \gamma\mathrm{tan}^{\mathrm{2}} \alpha+\mathrm{2tan}^{\mathrm{2}} \alpha\mathrm{tan}^{\mathrm{2}} \beta\mathrm{tan}^{\mathrm{2}} \gamma=\mathrm{1} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{sin}^{\mathrm{2}} \alpha+\mathrm{sin}^{\mathrm{2}} \beta+\mathrm{sin}^{\mathrm{2}} \gamma \\ $$

Question Number 143348 Answers: 1 Comments: 0

if ((cos^4 x)/(cos^2 y))+((sin^4 x)/(sin^2 y))=1 then find ((cos^4 y)/(cos^2 x))+((sin^4 y)/(sin^2 x))=?

$$\mathrm{if}\:\:\frac{\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{y}}+\frac{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{y}}=\mathrm{1}\:\mathrm{then}\: \\ $$$$\mathrm{find}\:\frac{\mathrm{cos}\:^{\mathrm{4}} \mathrm{y}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}+\frac{\mathrm{sin}\:^{\mathrm{4}} \mathrm{y}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}=? \\ $$

Question Number 143339 Answers: 1 Comments: 1

Question Number 143332 Answers: 2 Comments: 2

Question Number 143329 Answers: 1 Comments: 0

log_(2x+1) (x^2 +1)+log_(x+1) (3(√(2x+5))+18)=2x

$$\mathrm{log}_{\mathrm{2x}+\mathrm{1}} \left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)+\mathrm{log}_{\mathrm{x}+\mathrm{1}} \left(\mathrm{3}\sqrt{\mathrm{2x}+\mathrm{5}}+\mathrm{18}\right)=\mathrm{2x} \\ $$

Question Number 143328 Answers: 3 Comments: 0

x+(√(xy))+y=14 and x^2 +xy+y^2 =84 , find x and y

$${x}+\sqrt{{xy}}+{y}=\mathrm{14}\:\:{and}\: \\ $$$${x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}} =\mathrm{84}\:, \\ $$$${find}\:{x}\:{and}\:{y} \\ $$

Question Number 143327 Answers: 1 Comments: 0

1/.lim_(n→+∝) ((6+((6+((6+..........+(6)^(1/3) ))^(1/3) ))^(1/3) ))^(1/3) =?

$$\mathrm{1}/.\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\sqrt[{\mathrm{3}}]{\mathrm{6}+\sqrt[{\mathrm{3}}]{\mathrm{6}+\sqrt[{\mathrm{3}}]{\mathrm{6}+..........+\sqrt[{\mathrm{3}}]{\mathrm{6}}}}}=? \\ $$

Question Number 143326 Answers: 1 Comments: 0

L= lim_(n→+∝) (((2^3 −1)(3^3 −1)(4^3 −1)...(n^3 −1))/((2^3 +1)(3^3 +1)(4^3 +1)...(n^3 +1)))

$$\mathrm{L}=\:\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\left(\mathrm{2}^{\mathrm{3}} −\mathrm{1}\right)\left(\mathrm{3}^{\mathrm{3}} −\mathrm{1}\right)\left(\mathrm{4}^{\mathrm{3}} −\mathrm{1}\right)...\left(\mathrm{n}^{\mathrm{3}} −\mathrm{1}\right)}{\left(\mathrm{2}^{\mathrm{3}} +\mathrm{1}\right)\left(\mathrm{3}^{\mathrm{3}} +\mathrm{1}\right)\left(\mathrm{4}^{\mathrm{3}} +\mathrm{1}\right)...\left(\mathrm{n}^{\mathrm{3}} +\mathrm{1}\right)} \\ $$

Question Number 143324 Answers: 2 Comments: 0

(x^(2x^(−(1/5)) ) )^(−1) =(1/(25)) solve for x

$$\left(\boldsymbol{{x}}^{\mathrm{2}\boldsymbol{{x}}^{−\frac{\mathrm{1}}{\mathrm{5}}} } \right)^{−\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{25}} \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\:\boldsymbol{\mathrm{x}} \\ $$

Question Number 143322 Answers: 1 Comments: 0

Question Number 143320 Answers: 3 Comments: 0

prove that: tan^2 36° + tan^2 72° = 5

$${prove}\:{that}:\:\:{tan}^{\mathrm{2}} \mathrm{36}°\:+\:{tan}^{\mathrm{2}} \mathrm{72}°\:=\:\mathrm{5} \\ $$

Question Number 143317 Answers: 0 Comments: 1

Question Number 143312 Answers: 1 Comments: 0

∫_0 ^∞ ((sin^4 x)/x^4 )dx=(π/3)

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{\mathrm{4}} \mathrm{x}}{\mathrm{x}^{\mathrm{4}} }\mathrm{dx}=\frac{\pi}{\mathrm{3}} \\ $$

Pg 743 Pg 744 Pg 745 Pg 746 Pg 747 Pg 748 Pg 749 Pg 750 Pg 751 Pg 752

Contact: info@tinkutara.com