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AlgebraQuestion and Answers: Page 62

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 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 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 198252 Answers: 1 Comments: 2

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 198197 Answers: 1 Comments: 1

please helpe sinz = 2. Find z

$${please}\:{helpe} \\ $$$${sinz}\:=\:\mathrm{2}.\:{Find}\:{z} \\ $$

Question Number 198175 Answers: 1 Comments: 0

Prove The following Functional equation: ζ(x,s)=((2Γ(1−s))/((2π)^((1−s)) )){sin(((πs)/2))Σ_(m=1) ^∞ [((cos(2πmx))/m^((1−s)) )]+cos(((πs)/2))Σ_(m=1) ^∞ [((sin(2πmx))/m^((1−s)) )]}

$${Prove}\:{The}\:{following}\:{Functional}\:{equation}: \\ $$$$\zeta\left({x},{s}\right)=\frac{\mathrm{2}\Gamma\left(\mathrm{1}−{s}\right)}{\left(\mathrm{2}\pi\right)^{\left(\mathrm{1}−{s}\right)} }\left\{{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{cos}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]+{cos}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{sin}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]\right\} \\ $$

Question Number 198166 Answers: 3 Comments: 0

if f(x)=x^2 +bx+c f(f(1))=f(f(2))=0 and f(1)≠f(2) find f(0)=?

$${if}\:{f}\left({x}\right)={x}^{\mathrm{2}} +{bx}+{c} \\ $$$${f}\left({f}\left(\mathrm{1}\right)\right)={f}\left({f}\left(\mathrm{2}\right)\right)=\mathrm{0}\:{and}\:{f}\left(\mathrm{1}\right)\neq{f}\left(\mathrm{2}\right) \\ $$$${find}\:{f}\left(\mathrm{0}\right)=? \\ $$

Question Number 198147 Answers: 1 Comments: 0

if a,x,y,b is an AP and a,p,q,b is a GP. prove that xy≥pq. (with a, b >0)

$${if}\:{a},{x},{y},{b}\:{is}\:{an}\:{AP}\:{and}\:{a},{p},{q},{b}\:{is}\:{a}\:{GP}. \\ $$$${prove}\:{that}\:{xy}\geqslant{pq}. \\ $$$$\left({with}\:{a},\:{b}\:>\mathrm{0}\right) \\ $$

Question Number 198132 Answers: 1 Comments: 0

Solve: ((log(x^2 +7x−5))/(log(x+2)))=2

$${Solve}: \\ $$$$\frac{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{7}\boldsymbol{\mathrm{x}}−\mathrm{5}\right)}{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}+\mathrm{2}\right)}=\mathrm{2} \\ $$

Question Number 198131 Answers: 1 Comments: 0

Resoudre log(x−3)+log(x−2)=log(x^2 −4x−21)