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Question Number 198313    Answers: 0   Comments: 1

Question Number 198311    Answers: 0   Comments: 0

Let {x_r }_(r=1) ^n be n positive real numbers Show That: (x_1 /(1+x_1 ^2 ))+(x_2 /(1+x_1 ^2 +x_2 ^2 ))+...+(x_n /(1+x_1 ^2 +x_2 ^2 +...+x_n ^2 ))<(√n)

$${Let}\:\left\{{x}_{{r}} \right\}_{{r}=\mathrm{1}} ^{{n}} {be}\:{n}\:{positive}\:{real}\:{numbers}\:{Show}\:{That}: \\ $$$$\frac{{x}_{\mathrm{1}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{{x}_{\mathrm{2}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} }+...+\frac{{x}_{{n}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} }<\sqrt{{n}} \\ $$

Question Number 198309    Answers: 0   Comments: 2

Question Number 198304    Answers: 0   Comments: 7

for {a_n } be a sequence of positive real numbers such that a_1 =1 , a_(n+1) ^2 −2a_n a_(n+1) −a_n = 0 , ∀ n≥ 1 than the sum of series Σ_(n=1) ^∞ (a_n /3^(n ) ) lies in the interval (A) (1,2] (B) (2,3] (C) (3,4] (D) (4,5]

$$\:\:\:\mathrm{for}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}\:\mathrm{be}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\:\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{a}_{\mathrm{1}} =\mathrm{1}\:,\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} −\mathrm{2a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} −\mathrm{a}_{\mathrm{n}} \:=\:\mathrm{0}\:,\:\forall\:\mathrm{n}\geqslant\:\mathrm{1} \\ $$$$\:\:\:\mathrm{than}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{series}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{a}_{\mathrm{n}} }{\mathrm{3}^{\mathrm{n}\:} }\:\:\mathrm{lies}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\:\:\left({A}\right)\:\:\left(\mathrm{1},\mathrm{2}\right]\:\:\:\:\left({B}\right)\:\:\left(\mathrm{2},\mathrm{3}\right]\:\:\:\:\left({C}\right)\:\:\left(\mathrm{3},\mathrm{4}\right]\:\:\:\:\left({D}\right)\:\:\left(\mathrm{4},\mathrm{5}\right] \\ $$$$\:\:\:\: \\ $$

Question Number 198302    Answers: 1   Comments: 0

Question Number 198296    Answers: 0   Comments: 0

Let {x_r }_(r=1) ^n be n positive real numbers Show That: (x_1 /(1+x_1 ^2 ))+(x_2 /(1+x_1 ^2 +x_2 ^2 ))+...+(x_n /(1+x_1 ^2 +x_2 ^2 +...+x_n ^2 ))<(√n)

$${Let}\:\left\{{x}_{{r}} \right\}_{{r}=\mathrm{1}} ^{{n}} {be}\:{n}\:{positive}\:{real}\:{numbers}\:{Show}\:{That}: \\ $$$$\frac{{x}_{\mathrm{1}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{{x}_{\mathrm{2}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} }+...+\frac{{x}_{{n}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} }<\sqrt{{n}} \\ $$

Question Number 198295    Answers: 1   Comments: 0

x^3 −((81x−8))^(1/3) = 2x^2 −(4/3)x+2

$$\:\:\:\mathrm{x}^{\mathrm{3}} −\sqrt[{\mathrm{3}}]{\mathrm{81x}−\mathrm{8}}\:=\:\mathrm{2x}^{\mathrm{2}} −\frac{\mathrm{4}}{\mathrm{3}}\mathrm{x}+\mathrm{2}\: \\ $$

Question Number 198293    Answers: 1   Comments: 0

Question Number 198267    Answers: 3   Comments: 0

Find the real values of n: n^6 −n^3 =2

$${Find}\:{the}\:{real}\:{values}\:{of}\:{n}:\:{n}^{\mathrm{6}} −{n}^{\mathrm{3}} =\mathrm{2} \\ $$

Question Number 198266    Answers: 1   Comments: 0

Question Number 198263    Answers: 1   Comments: 0

Question Number 198260    Answers: 0   Comments: 0

help ! (i^→ ,j^→ ,k^→ ) est une base orthonormee. A, B, C et D sont des points de l′espace tels que : AB^(→) =i^→ +j^→ +k^→ AC^(→) =2i^→ +3j^→ +k^→ AD^(→) =i^→ −2j^→ +2k^→ . Determine tous les points P tels que (DP) ⊂ (OAB) et AP^(→) soit un vecteur unitaire orthogonal a AD^(→) . by axel

$$\mathrm{help}\:! \\ $$$$\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}},\overset{\rightarrow} {{k}}\right)\:\mathrm{est}\:\mathrm{une}\:\mathrm{base}\:\mathrm{orthonormee}. \\ $$$$\mathrm{A},\:\mathrm{B},\:\mathrm{C}\:\mathrm{et}\:\mathrm{D}\:\mathrm{sont}\:\mathrm{des}\:\mathrm{points}\:\mathrm{de}\:\mathrm{l}'\mathrm{espace} \\ $$$$\mathrm{tels}\:\mathrm{que}\::\: \\ $$$$\overset{\rightarrow} {\mathrm{AB}}=\overset{\rightarrow} {{i}}+\overset{\rightarrow} {{j}}+\overset{\rightarrow} {{k}} \\ $$$$\overset{\rightarrow} {\mathrm{AC}}=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{3}\overset{\rightarrow} {{j}}+\overset{\rightarrow} {{k}} \\ $$$$\overset{\rightarrow} {\mathrm{AD}}=\overset{\rightarrow} {{i}}−\mathrm{2}\overset{\rightarrow} {{j}}+\mathrm{2}\overset{\rightarrow} {{k}}. \\ $$$$\boldsymbol{\mathrm{Determine}}\:\boldsymbol{\mathrm{tous}}\:\boldsymbol{\mathrm{les}}\:\boldsymbol{\mathrm{points}}\:\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{tels}}\:\boldsymbol{\mathrm{que}}\: \\ $$$$\left(\boldsymbol{\mathrm{DP}}\right)\:\subset\:\left(\boldsymbol{\mathrm{OAB}}\right)\:\boldsymbol{\mathrm{et}}\:\overset{\rightarrow} {\boldsymbol{\mathrm{AP}}}\:\boldsymbol{\mathrm{soit}}\:\boldsymbol{\mathrm{un}}\:\boldsymbol{\mathrm{vecteur}}\: \\ $$$$\boldsymbol{\mathrm{unitaire}}\:\boldsymbol{\mathrm{orthogonal}}\:\boldsymbol{\mathrm{a}}\:\overset{\rightarrow} {\boldsymbol{\mathrm{AD}}}. \\ $$$${by}\:{axel} \\ $$

Question Number 198252    Answers: 1   Comments: 2

Question Number 198249    Answers: 1   Comments: 4

if sin(x+ϕ)+cos2x≤(√(3 )) ϕ=?

$${if}\:\:{sin}\left({x}+\varphi\right)+{cos}\mathrm{2}{x}\leqslant\sqrt{\mathrm{3}\:}\: \\ $$$$\varphi=? \\ $$

Question Number 198246    Answers: 0   Comments: 0

prove that Σ_(i=1) ^(2n−1) (((−1)^(i−1) )/i)>ln2+(1/(4n))

$${prove}\:{that} \\ $$$$\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{i}−\mathrm{1}} }{{i}}>{ln}\mathrm{2}+\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$

Question Number 198244    Answers: 0   Comments: 0

Question Number 198243    Answers: 3   Comments: 0

find the sum of the first n terms from 1, 2+3, 4+5+6, 7+8+9+10, ...

$${find}\:{the}\:{sum}\:{of}\:{the}\:{first}\:{n}\:{terms}\:{from} \\ $$$$\mathrm{1},\:\mathrm{2}+\mathrm{3},\:\mathrm{4}+\mathrm{5}+\mathrm{6},\:\mathrm{7}+\mathrm{8}+\mathrm{9}+\mathrm{10},\:... \\ $$

Question Number 198242    Answers: 1   Comments: 3

How many numbers with a maximum of 5 digits, greater than 4000, can be formed with the digits 2, 3, 4, 5, 6; if repetition is allowed for 2 and 3 only?

How many numbers with a maximum of 5 digits, greater than 4000, can be formed with the digits 2, 3, 4, 5, 6; if repetition is allowed for 2 and 3 only?

Question Number 198241    Answers: 1   Comments: 0

Question Number 198237    Answers: 1   Comments: 0

if −(√3)≤sin(x+ϕ)+cosx≤(√3) ϕ=?

$${if}\:\:−\sqrt{\mathrm{3}}\leqslant{sin}\left({x}+\varphi\right)+{cosx}\leqslant\sqrt{\mathrm{3}} \\ $$$$\varphi=? \\ $$

Question Number 198235    Answers: 1   Comments: 0

Question Number 198269    Answers: 1   Comments: 0

calcul Σ_(k=o) ^n sin(k)

$${calcul} \\ $$$$\underset{{k}={o}} {\overset{{n}} {\sum}}{sin}\left({k}\right) \\ $$

Question Number 198232    Answers: 1   Comments: 0

find Σ_(k=o) ^n sin(k)

$${find}\: \\ $$$$\underset{{k}={o}} {\overset{{n}} {\sum}}{sin}\left({k}\right) \\ $$

Question Number 198231    Answers: 1   Comments: 0

Question Number 198228    Answers: 1   Comments: 0

(4x^2 +2x+1)^(x^2 −x) >1

$$\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right)^{{x}^{\mathrm{2}} −{x}} >\mathrm{1} \\ $$

Question Number 198222    Answers: 1   Comments: 1

log _4 (5^x −3^x ) = log _5 (4^x +3^(x ) )

$$\:\:\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \right) \\ $$

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