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Question Number 181217 Answers: 1 Comments: 0

Question Number 181207 Answers: 1 Comments: 0

cacul ∀x∈]0,1[ Σ_(n=0) ^(+∞) nx^n

$${cacul} \\ $$$$\left.\forall{x}\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$$\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}{nx}^{{n}} \\ $$

Question Number 181201 Answers: 3 Comments: 0

Question Number 181183 Answers: 1 Comments: 0

Ω_n = determinant ((1,1,1,(...),1),(1,2^2 ,2^3 ,(...),2^n ),(1,3^2 ,3^3 ,(...),3^n ),((...),(...),(...),(...),(...)),(1,n^2 ,n^3 ,(...),n^n )) , n ∈ N^∗ Find: Ω =lim_(n→∞) ((Ω_(n+1) /Ω_n ))^(1/n)

Question Number 181182 Answers: 1 Comments: 0

For what values of a does the system of equations only have one solution: { ((a(x^4 +1)=y+2−∣x∣)),((x^2 +y^2 =4)) :}

$${For}\:{what}\:{values}\:{of}\:{a}\:{does}\:{the}\:{system} \\ $$$${of}\:{equations}\:{only}\:{have}\:{one}\:{solution}: \\ $$$$\begin{cases}{{a}\left({x}^{\mathrm{4}} +\mathrm{1}\right)={y}+\mathrm{2}−\mid{x}\mid}\\{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{4}}\end{cases} \\ $$

Question Number 181173 Answers: 0 Comments: 1

Question Number 181172 Answers: 0 Comments: 2

write snell′s law that light move a concentrative medium to nonconcentrative medium and show with a shape.

$${write}\:{snell}'{s}\:{law}\:{that}\:{light}\:{move}\:{a}\: \\ $$$${concentrative}\:{medium}\:{to}\:{nonconcentrative} \\ $$$${medium}\:{and}\:{show}\:{with}\:{a}\:{shape}. \\ $$

Question Number 181171 Answers: 0 Comments: 1

prove snell′s law

$${prove}\:{snell}'{s}\:{law} \\ $$

Question Number 181152 Answers: 0 Comments: 0

solve the integral ∫_0 ^( 1) ((ln(1+x)ln(1−x))/(1+x))dx=???

$$\boldsymbol{{solve}}\:\boldsymbol{{the}}\:\boldsymbol{{integral}} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\boldsymbol{{ln}}\left(\mathrm{1}+\boldsymbol{{x}}\right)\boldsymbol{{ln}}\left(\mathrm{1}−\boldsymbol{{x}}\right)}{\mathrm{1}+\boldsymbol{{x}}}\boldsymbol{{dx}}=??? \\ $$

Question Number 181151 Answers: 4 Comments: 0

Question Number 181196 Answers: 0 Comments: 0

Let a_1 , a_2 , a_3 , ...a_(2022) be numbers ranging from (0, +∞) \ {1}, for which the function f : R→R is defined as f(x)=a_1 ^x +a_2 ^x +a_3 ^x +...a_(2022) ^x . If f(2022)=f(−2022)=2022 prove that this function is constant.

$${Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:...{a}_{\mathrm{2022}} \:{be}\:{numbers} \\ $$$${ranging}\:{from}\:\left(\mathrm{0},\:+\infty\right)\:\backslash\:\left\{\mathrm{1}\right\},\:{for}\:{which} \\ $$$${the}\:{function}\:{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:{is}\:{defined}\:{as} \\ $$$${f}\left({x}\right)={a}_{\mathrm{1}} ^{{x}} +{a}_{\mathrm{2}} ^{{x}} +{a}_{\mathrm{3}} ^{{x}} +...{a}_{\mathrm{2022}} ^{{x}} . \\ $$$${If}\:{f}\left(\mathrm{2022}\right)={f}\left(−\mathrm{2022}\right)=\mathrm{2022}\:{prove} \\ $$$${that}\:{this}\:{function}\:{is}\:{constant}. \\ $$

Question Number 181140 Answers: 0 Comments: 6

I watch a favorite TV program daily for 30 min. If there were ads every 3 hours, what′s the probability that i will see ads once again the next day while watching that program?

$${I}\:{watch}\:{a}\:{favorite}\:{TV}\:{program}\:{daily}\:{for}\:\mathrm{30}\:{min}. \\ $$$$\:{If}\:{there}\:{were}\:{ads}\:{every}\:\mathrm{3}\:{hours},\:{what}'{s}\:{the} \\ $$$$\:{probability}\:{that}\:{i}\:{will}\:{see}\:{ads}\:{once}\:{again}\:{the}\:{next} \\ $$$$\:{day}\:{while}\:{watching}\:{that}\:{program}? \\ $$$$ \\ $$

Question Number 181138 Answers: 1 Comments: 0

If a + b + c = 0 then, (1/(2a^2 + bc)) + (1/(2b^2 + ca)) + (1/(2c^2 + ab)) = ?

$$\mathrm{If}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{then}, \\ $$$$\frac{\mathrm{1}}{\mathrm{2}{a}^{\mathrm{2}} \:+\:{bc}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{b}^{\mathrm{2}} \:+\:{ca}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{c}^{\mathrm{2}} \:+\:{ab}}\:=\:? \\ $$

Question Number 181153 Answers: 2 Comments: 0

Question Number 181130 Answers: 0 Comments: 0

Question Number 181128 Answers: 0 Comments: 2

Question Number 181154 Answers: 1 Comments: 0

(log_(15) 5)^2 +(log_(15) 3)(log_(15) 75)=?

$$\left({log}_{\mathrm{15}} \mathrm{5}\right)^{\mathrm{2}} +\left({log}_{\mathrm{15}} \mathrm{3}\right)\left({log}_{\mathrm{15}} \mathrm{75}\right)=? \\ $$$$ \\ $$

Question Number 181125 Answers: 1 Comments: 3

Let the acute triangle ΔABC have an outer circumscribed circle, whose tangents at the points B and C intersect at point P. Let D and E be the projections of perpendicular lines from point P on AC and AB. Prove that the interdection point of the heights of ΔADE is the midpoint of BC

$${Let}\:{the}\:{acute}\:{triangle}\:\Delta{ABC}\:\:{have} \\ $$$${an}\:{outer}\:{circumscribed}\:{circle}, \\ $$$${whose}\:{tangents}\:{at}\:{the}\:{points}\:{B}\:{and}\:{C} \\ $$$${intersect}\:{at}\:{point}\:{P}.\:{Let}\:{D}\:{and}\:{E}\:{be} \\ $$$${the}\:{projections}\:{of}\:{perpendicular} \\ $$$${lines}\:{from}\:{point}\:{P}\:{on}\:{AC}\:{and}\:{AB}. \\ $$$${Prove}\:{that}\:{the}\:{interdection}\:{point}\:{of} \\ $$$${the}\:{heights}\:{of}\:\Delta{ADE}\:{is}\:{the}\:{midpoint} \\ $$$${of}\:{BC} \\ $$

Question Number 181144 Answers: 3 Comments: 0

Solve for x : (((3x − 28)/(3x − 26)))^3 = ((x − 10)/(x − 8))

$$\mathrm{Solve}\:\mathrm{for}\:{x}\:: \\ $$$$\left(\frac{\mathrm{3}{x}\:−\:\mathrm{28}}{\mathrm{3}{x}\:−\:\mathrm{26}}\right)^{\mathrm{3}} \:=\:\frac{{x}\:−\:\mathrm{10}}{{x}\:−\:\mathrm{8}} \\ $$

Question Number 181104 Answers: 2 Comments: 1

lim_(x→∞) ((ln (1+(4/x)))/(π−arctan (2x))) =?

$$\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{4}}{\mathrm{x}}\right)}{\pi−\mathrm{arctan}\:\left(\mathrm{2x}\right)}\:=?\: \\ $$

Question Number 181099 Answers: 2 Comments: 0

prove that for every positivenumber p e q wee have: p+q≥(√(4pq))

$$ \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\: \\ $$$$\mathrm{positivenumber}\:\mathrm{p}\:\mathrm{e}\:\mathrm{q}\:\mathrm{wee} \\ $$$$\mathrm{hav}{e}: \\ $$$${p}+{q}\geqslant\sqrt{\mathrm{4}{pq}} \\ $$

Question Number 181089 Answers: 1 Comments: 0