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Question Number 195368 Answers: 0 Comments: 0

Question Number 195365 Answers: 2 Comments: 0

_(x→y) ^(lim) ((tan x−tany)/(x−y))

$$\:\: \\ $$$$\:\underset{\mathrm{x}\rightarrow\mathrm{y}} {\overset{\mathrm{lim}} {\:}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{tany}}{\mathrm{x}−\mathrm{y}} \\ $$$$ \\ $$

Question Number 195364 Answers: 1 Comments: 0

show that for any natural number n, the natural number (3−(√5))^n +(3+(√5))^n is divisible by 2^n .

$$ \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{natural}\:\mathrm{number}\:{n},\: \\ $$$$\mathrm{the}\:\mathrm{natural}\:\mathrm{number}\:\left(\mathrm{3}−\sqrt{\mathrm{5}}\right)^{{n}} +\left(\mathrm{3}+\sqrt{\mathrm{5}}\right)^{{n}} \:\mathrm{is}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{2}^{{n}} . \\ $$

Question Number 195361 Answers: 0 Comments: 0

Question Number 195357 Answers: 2 Comments: 0

Question Number 195356 Answers: 2 Comments: 0

remark to question 195301 and similar ones x^2 +y=a x+y^2 =b a, b >0 how many solutions depending on a, b?

$$\mathrm{remark}\:\mathrm{to}\:\mathrm{question}\:\mathrm{195301}\:\mathrm{and}\:\mathrm{similar}\:\mathrm{ones} \\ $$$${x}^{\mathrm{2}} +{y}={a} \\ $$$${x}+{y}^{\mathrm{2}} ={b} \\ $$$${a},\:{b}\:>\mathrm{0} \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{solutions}\:\mathrm{depending}\:\mathrm{on}\:{a},\:{b}? \\ $$

Question Number 195352 Answers: 0 Comments: 0

Question Number 195349 Answers: 0 Comments: 0

Question Number 195344 Answers: 2 Comments: 0

Question Number 195393 Answers: 1 Comments: 0

prove that lim_(x→0) (((Σ_(k=1) ^n (1−(1/(2k)))^x )/n))^(1/( x )) = (1/4)(C_(2n) ^n )^(1/n)

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{\:\:\boldsymbol{{x}}\:\:}]{\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{k}}\right)^{{x}} }{{n}}}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\sqrt[{\boldsymbol{{n}}}]{\mathrm{C}_{\mathrm{2}\boldsymbol{\mathrm{n}}} ^{\boldsymbol{\mathrm{n}}} } \\ $$

Question Number 195395 Answers: 1 Comments: 0

Question Number 197578 Answers: 1 Comments: 1

Question Number 200284 Answers: 1 Comments: 0

Prove that for any set A containing n elements, ∣P(A)∣=2^n .

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{set}\:{A}\:\mathrm{containing}\:{n} \\ $$$$\mathrm{elements},\:\mid\mathcal{P}\left({A}\right)\mid=\mathrm{2}^{{n}} . \\ $$

Question Number 195331 Answers: 1 Comments: 0

Question Number 195330 Answers: 0 Comments: 1

Question Number 195325 Answers: 2 Comments: 0

prove that lim_(x→(π/2)) ((tan((x/2))−1)/(x−(π/2)))=1

$${prove}\:{that} \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{{tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}}{{x}−\frac{\pi}{\mathrm{2}}}=\mathrm{1} \\ $$

Question Number 200302 Answers: 1 Comments: 0

Question Number 195320 Answers: 1 Comments: 0

I_n =∫_0 ^( +∞) t^(−2t) sin^(2n) tdt Prove that I_n =(1/(1−e^(−2π) )) ∫^( π) _( 0) e^(−2t) sin^(2n) t dt and I_n ∽ _(∞) (1/(2sh(π)))(√(π/n))

$$\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\:+\infty} {t}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {tdt} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{I}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{2}\pi} }\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi} {e}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {t}\:{dt} \\ $$$$\mathrm{and}\:\:\mathrm{I}_{\mathrm{n}} \underset{\infty} {\:\:\backsim\:\:}\:\frac{\mathrm{1}}{\mathrm{2}{sh}\left(\pi\right)}\sqrt{\frac{\pi}{{n}}} \\ $$

Question Number 195342 Answers: 4 Comments: 0

1. Prove that ∀n ∈ N^∗ , 4^n (n!)^3 < (n+1)^(3n) . 2. Solve the equations in Z^2 : a./ 2x^3 +xy−7=0 , b./ x(x+1)(x+7)(x+8)=y^2 .

$$\:\:\mathrm{1}.\:\mathrm{Prove}\:\mathrm{that}\:\:\forall{n}\:\in\:\mathbb{N}^{\ast} \:,\:\mathrm{4}^{{n}} \left({n}!\right)^{\mathrm{3}} \:<\:\left({n}+\mathrm{1}\right)^{\mathrm{3}{n}} \:. \\ $$$$\mathrm{2}.\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{in}\:\mathbb{Z}^{\mathrm{2}} \:: \\ $$$$\:\:\:\:\:{a}./\:\:\mathrm{2}{x}^{\mathrm{3}} +{xy}−\mathrm{7}=\mathrm{0}\:, \\ $$$$\:\:\:\:\:{b}./\:\:{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{7}\right)\left({x}+\mathrm{8}\right)={y}^{\mathrm{2}} . \\ $$

Question Number 195341 Answers: 0 Comments: 0

Question Number 195315 Answers: 1 Comments: 0

lim_(x→0) ((tan (a−x) tan (a+x)−tan^2 a)/(2x^2 ))

$$\:\:\:\:\: \underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left({a}−{x}\right)\:\mathrm{tan}\:\left({a}+{x}\right)−\mathrm{tan}\:^{\mathrm{2}} {a}}{\mathrm{2}{x}^{\mathrm{2}} }\:\:\:\:\:\:\: \\ $$

Question Number 195555 Answers: 1 Comments: 0

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Question Number 195553 Answers: 1 Comments: 0

Question Number 195301 Answers: 1 Comments: 0

{ ((x^2 +y=11)),((x+y^2 =7)) :}⇒ x,y=?

$$\begin{cases}{{x}^{\mathrm{2}} +{y}=\mathrm{11}}\\{{x}+{y}^{\mathrm{2}} =\mathrm{7}}\end{cases}\Rightarrow\:{x},{y}=? \\ $$

Question Number 195292 Answers: 1 Comments: 0

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