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AlgebraQuestion and Answers: Page 249

Question Number 99117 Answers: 1 Comments: 0

let a,b,c ∈R determine the minimum value ((3a)/(b+c))+((4b)/(a+c))+((5c)/(a+b))

$${let}\:{a},{b},{c}\:\in\mathbb{R}\:{determine}\:{the}\:{minimum} \\ $$$${value} \\ $$$$ \\ $$$$\frac{\mathrm{3}{a}}{{b}+{c}}+\frac{\mathrm{4}{b}}{{a}+{c}}+\frac{\mathrm{5}{c}}{{a}+{b}} \\ $$

Question Number 99094 Answers: 1 Comments: 0

find ((9+9((9+9((9+9((9+...))^(1/(3 )) ))^(1/(3 )) ))^(1/(3 )) ))^(1/(3 )) − (√(8−(√(8−(√(8+(√(8−(√(8−(√(8−(√(8−(√)))))))...))))))))

$$\mathrm{find}\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9}\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9}\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9}\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+...}}}}− \\ $$$$\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}+\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{}}}}...}}}}\: \\ $$

Question Number 99089 Answers: 1 Comments: 0

((9+((9+((9+((9+...))^(1/(3 )) ))^(1/(3 )) ))^(1/(3 )) ))^(1/(3 )) −(√(8−(√(8−(√(8−(√(8−...))))))))

$$\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+...}}}}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−...}}}} \\ $$

Question Number 99045 Answers: 1 Comments: 0

{ ((x^2 +y^2 = 10)),((x^2 −5xy+6y^2 = 0)) :} find x &y

$$\begin{cases}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{10}}\\{\mathrm{x}^{\mathrm{2}} −\mathrm{5xy}+\mathrm{6y}^{\mathrm{2}} \:=\:\mathrm{0}}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{x}\:\&\mathrm{y}\: \\ $$

Question Number 99005 Answers: 2 Comments: 0

Σ_(m = 1) ^∞ Σ_(n = 1) ^∞ (1/(mn(m+n))) ?

$$\underset{\mathrm{m}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{mn}\left(\mathrm{m}+\mathrm{n}\right)}\:?\: \\ $$

Question Number 98983 Answers: 0 Comments: 2

let a,b,c be positive real numbers such that ab+bc+ac=3 prove the inquality ((a(b^2 +c^2 ))/(a^2 +bc))+((b(c^2 +a^2 ))/(b^2 +ac))+((c(b^2 +a^2 ))/(c^2 +ab))≥3

$${let}\:{a},{b},{c}\:{be}\:{positive}\:{real}\:{numbers}\:{such} \\ $$$${that}\:{ab}+{bc}+{ac}=\mathrm{3}\: \\ $$$${prove}\:{the}\:{inquality} \\ $$$$ \\ $$$$\frac{{a}\left({b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)}{{a}^{\mathrm{2}} +{bc}}+\frac{{b}\left({c}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)}{{b}^{\mathrm{2}} +{ac}}+\frac{{c}\left({b}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)}{{c}^{\mathrm{2}} +{ab}}\geqslant\mathrm{3} \\ $$

Question Number 98925 Answers: 1 Comments: 1

If the curve shown below has the equation, y=(x−p)(x^3 −bx−c) then find q/p in terms of b and c.

$${If}\:{the}\:{curve}\:{shown}\:{below}\:{has}\:{the}\: \\ $$$${equation},\:\:{y}=\left({x}−{p}\right)\left({x}^{\mathrm{3}} −{bx}−{c}\right) \\ $$$${then}\:{find}\:\:{q}/{p}\:\:{in}\:{terms}\:{of}\:{b}\:{and}\:{c}. \\ $$

Question Number 98914 Answers: 3 Comments: 2

2x+3y=5 (x^2 +y^2 )_(min) =?

$$\mathrm{2}{x}+\mathrm{3}{y}=\mathrm{5} \\ $$$$\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \underset{{min}} {\right)}=? \\ $$

Question Number 98859 Answers: 2 Comments: 1

Question Number 98857 Answers: 0 Comments: 0

Question Number 98844 Answers: 3 Comments: 0

Please explain: Σ_(1 ≤ i < j ≤ n) ij = Σ_(j = 2) ^n ((j(j − 1)j)/2) I want to know how L.H.S = R.H.S

Question Number 98842 Answers: 2 Comments: 1

let f(x) be a dolvnomial of degree 4 such that f(1)=1 , f(2)=2 ,f(3)=3,f(4)=4 then f(6)=?

$${let}\:{f}\left({x}\right)\:{be}\:{a}\:{dolvnomial}\:{of}\:{degree}\:\mathrm{4}\: \\ $$$${such}\:{that}\:{f}\left(\mathrm{1}\right)=\mathrm{1}\:,\:{f}\left(\mathrm{2}\right)=\mathrm{2}\:,{f}\left(\mathrm{3}\right)=\mathrm{3},{f}\left(\mathrm{4}\right)=\mathrm{4} \\ $$$${then}\:{f}\left(\mathrm{6}\right)=? \\ $$

Question Number 98818 Answers: 0 Comments: 0

Question Number 98770 Answers: 0 Comments: 2

Question Number 98761 Answers: 0 Comments: 5

(√(x+(√x) )) −(√(x−(√x))) = m(√(x/(x+(√x)))) m is a real parameter

$$\sqrt{\mathrm{x}+\sqrt{\mathrm{x}}\:}\:−\sqrt{\mathrm{x}−\sqrt{\mathrm{x}}}\:=\:\mathrm{m}\sqrt{\frac{\mathrm{x}}{\mathrm{x}+\sqrt{\mathrm{x}}}} \\ $$$$\mathrm{m}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{parameter} \\ $$

Question Number 98750 Answers: 0 Comments: 0

Question Number 98796 Answers: 2 Comments: 0

Question Number 98570 Answers: 1 Comments: 1

a,b,c>0 prove: (a/(√(a^2 +8bc)))+(b/(√(b^2 +8ac)))+(c/(√(c^2 +8ab)))≥1 help please...

$$\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}}>\mathrm{0}\:\:\:\:\:\:\:\boldsymbol{{prove}}: \\ $$$$\frac{\boldsymbol{{a}}}{\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{bc}}}}+\frac{\boldsymbol{{b}}}{\sqrt{\boldsymbol{{b}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{ac}}}}+\frac{\boldsymbol{{c}}}{\sqrt{\boldsymbol{{c}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{ab}}}}\geqslant\mathrm{1} \\ $$$$\boldsymbol{{help}}\:\boldsymbol{{please}}... \\ $$

Question Number 98521 Answers: 1 Comments: 0

Question Number 98613 Answers: 0 Comments: 0

Question Number 98448 Answers: 1 Comments: 5

6^(273) +8^(273) :49 prove the division

$$\:\:\:\:\:\:\:\mathrm{6}^{\mathrm{273}} +\mathrm{8}^{\mathrm{273}} \:\::\mathrm{49}\:\:\:\boldsymbol{{prove}}\:\:\boldsymbol{{the}}\:\:\boldsymbol{{divi}\mathrm{s}{ion}} \\ $$

Question Number 98416 Answers: 3 Comments: 0

Question Number 98280 Answers: 1 Comments: 0

Let {a_n } be a sequence such that a_1 = 2, a_(n + 1) = ((3a_n + 4)/(2a_n + 3)), n ≥ 1, find a_n

$$\boldsymbol{\mathrm{Let}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{be}}\:\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{sequence}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\:\:\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:=\:\:\mathrm{2}, \\ $$$$\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}} \:\:=\:\:\frac{\mathrm{3}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:+\:\:\mathrm{4}}{\mathrm{2}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:\:+\:\:\mathrm{3}},\:\:\:\:\:\boldsymbol{\mathrm{n}}\:\geqslant\:\mathrm{1},\:\:\:\:\:\boldsymbol{\mathrm{find}}\:\:\:\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \\ $$

Question Number 98268 Answers: 0 Comments: 7

let p(x) be a polynomial function of (n−1)^(th) degree and p(k)=k for k=1,2,3,...,n find p(0) and p(n+1). example: n=10

$${let}\:{p}\left({x}\right)\:{be}\:{a}\:{polynomial}\:{function}\:{of} \\ $$$$\left({n}−\mathrm{1}\right)^{{th}} \:{degree}\:{and} \\ $$$${p}\left({k}\right)={k}\:{for}\:{k}=\mathrm{1},\mathrm{2},\mathrm{3},...,{n} \\ $$$${find}\:{p}\left(\mathrm{0}\right)\:{and}\:{p}\left({n}+\mathrm{1}\right). \\ $$$${example}:\:{n}=\mathrm{10} \\ $$

Question Number 98267 Answers: 2 Comments: 0

∀ a,b>0 , a^2 +b^2 =1 prove that ((1/a)+(1/b))((b/(a^2 +1))+(a/(b^2 +1)))≥(8/3)

$$\forall\:{a},{b}>\mathrm{0}\:,\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{1} \\ $$$${prove}\:{that} \\ $$$$\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\left(\frac{{b}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{a}}{{b}^{\mathrm{2}} +\mathrm{1}}\right)\geqslant\frac{\mathrm{8}}{\mathrm{3}} \\ $$

Question Number 98215 Answers: 3 Comments: 2

Find the nth term of the sequence {a_n } such that ((a_1 + a_2 + ... + a_n )/n) = n + (1/n) (n = 1, 2, 3, ...)

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