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AlgebraQuestion and Answers: Page 29

Question Number 209929 Answers: 0 Comments: 0

Question Number 209918 Answers: 1 Comments: 0

Find: ∫_0 ^( ∞) (({x}^([x]) )/([x] + 1)) dx = ? {x} → fractional part [x] → full part

$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\left\{\mathrm{x}\right\}^{\left[\boldsymbol{\mathrm{x}}\right]} }{\left[\mathrm{x}\right]\:+\:\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$$$\left\{\mathrm{x}\right\}\:\rightarrow\:\mathrm{fractional}\:\mathrm{part} \\ $$$$\left[\mathrm{x}\right]\:\:\:\rightarrow\:\mathrm{full}\:\mathrm{part} \\ $$

Question Number 209876 Answers: 1 Comments: 0

1,6 = (((2x)^2 )/((3−2x)^2 ∙ (2−x))) find: x = ?

$$\mathrm{1},\mathrm{6}\:\:=\:\:\frac{\left(\mathrm{2x}\right)^{\mathrm{2}} }{\left(\mathrm{3}−\mathrm{2x}\right)^{\mathrm{2}} \:\centerdot\:\left(\mathrm{2}−\mathrm{x}\right)}\:\:\:\mathrm{find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 209872 Answers: 1 Comments: 0

Solve for x∈R: x^3 −3x^2 +2=(√(x+1))

$$\mathrm{Solve}\:\mathrm{for}\:{x}\in\mathbb{R}: \\ $$$${x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}=\sqrt{{x}+\mathrm{1}} \\ $$

Question Number 209852 Answers: 1 Comments: 0

Find: Σ_(n=1) ^∞ arctan ((2/n^2 )) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{arctan}\:\left(\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} }\right)\:=\:? \\ $$

Question Number 209837 Answers: 2 Comments: 0

if the series Σ_(n=1) ^∞ (1/n^2 ) converges to k . find the convergence value of Σ_(n=1) ^∞ (1/((2n+1)^2 ))

$$ \\ $$$${if}\:{the}\:{series}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:{converges}\:{to}\:{k}\:.\:\:{find}\:\:{the}\:{convergence}\:{value}\:{of}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 209812 Answers: 0 Comments: 0

Question Number 209768 Answers: 1 Comments: 0

Express tan(3) in surd form

$$\boldsymbol{{Express}}\:\boldsymbol{{tan}}\left(\mathrm{3}\right)\:\boldsymbol{{in}}\:\boldsymbol{{surd}}\:\boldsymbol{{form}} \\ $$

Question Number 209735 Answers: 3 Comments: 0

tan(3x) + tan(5x) = 2 Find: x = ?

$$\mathrm{tan}\left(\mathrm{3x}\right)\:\:+\:\:\mathrm{tan}\left(\mathrm{5x}\right)\:\:=\:\:\mathrm{2} \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 209719 Answers: 1 Comments: 0

If: 7^(243) = a...bc^(−) Find: b∙c = ?

$$\mathrm{If}: \\ $$$$\mathrm{7}^{\mathrm{243}} \:\:=\:\:\overline {\mathrm{a}...\mathrm{bc}} \\ $$$$\mathrm{Find}: \\ $$$$\mathrm{b}\centerdot\mathrm{c}\:=\:? \\ $$

Question Number 209706 Answers: 2 Comments: 0

find the sum of sin^2 1°+sin^2 2°+...+sin^2 60°=?

$${find}\:{the}\:{sum}\:{of}\: \\ $$$$\mathrm{sin}^{\mathrm{2}} \:\mathrm{1}°+\mathrm{sin}^{\mathrm{2}} \:\mathrm{2}°+...+\mathrm{sin}^{\mathrm{2}} \:\mathrm{60}°=? \\ $$

Question Number 209695 Answers: 1 Comments: 0

If: (a + b)∙(√2) = 7∙(a−b−4) Find: (2a + b) = ?

$$\mathrm{If}: \\ $$$$\left(\mathrm{a}\:+\:\mathrm{b}\right)\centerdot\sqrt{\mathrm{2}}\:=\:\mathrm{7}\centerdot\left(\mathrm{a}−\mathrm{b}−\mathrm{4}\right) \\ $$$$\mathrm{Find}: \\ $$$$\left(\mathrm{2a}\:+\:\mathrm{b}\right)\:=\:? \\ $$

Question Number 209694 Answers: 1 Comments: 0

x^2 +xy+y^2 =α^2 y^2 +yz+z^2 =β^2 z^2 +zx+x^2 =α^2 +β^2 Find x+y+z for x, y, z ∈R^+

$${x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}} =\alpha^{\mathrm{2}} \\ $$$${y}^{\mathrm{2}} +{yz}+{z}^{\mathrm{2}} =\beta^{\mathrm{2}} \\ $$$${z}^{\mathrm{2}} +{zx}+{x}^{\mathrm{2}} =\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} \\ $$$$\mathrm{Find}\:{x}+{y}+{z}\:\mathrm{for}\:{x},\:{y},\:{z}\:\in\mathbb{R}^{+} \\ $$

Question Number 209633 Answers: 2 Comments: 0

Question Number 209580 Answers: 2 Comments: 0

If a_n >0 and lim_(n→∞) a_n = 0 Find: lim_(n→∞) (1/n) Σ_(k=1) ^n ln ((k/n) + a_n ) = ?

$$\mathrm{If}\:\:\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} >\mathrm{0}\:\:\:\mathrm{and}\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:=\:\mathrm{0} \\ $$$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{ln}\:\left(\frac{\mathrm{k}}{\mathrm{n}}\:+\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \right)\:=\:? \\ $$

Question Number 209542 Answers: 2 Comments: 0

∫(dx/(x^(15) −x^(11) ))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{{x}^{\mathrm{15}} −{x}^{\mathrm{11}} } \\ $$$$ \\ $$$$ \\ $$

Question Number 209539 Answers: 0 Comments: 0

Question Number 209531 Answers: 0 Comments: 1

(y^(′′) /y) = 4x^2 + 2

$$\frac{\mathrm{y}^{''} }{\mathrm{y}}\:\:=\:\:\mathrm{4x}^{\mathrm{2}} \:\:+\:\:\mathrm{2} \\ $$

Question Number 209514 Answers: 1 Comments: 0

If Σ a_n is absolutely convergent, prove that Σ (a_n /n) is also absolutely convergent.

$$\mathrm{If}\:\Sigma\:{a}_{{n}} \:\mathrm{is}\:\mathrm{absolutely}\:\mathrm{convergent},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\Sigma\:\frac{{a}_{{n}} }{{n}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{absolutely}\:\mathrm{convergent}. \\ $$

Question Number 209456 Answers: 2 Comments: 0

Question Number 209455 Answers: 1 Comments: 0

Question Number 209436 Answers: 1 Comments: 2

{ ((x + y + z = 1)),((42x + 44y + 30z = 42)) :} (x,y,z)=(1,0,0) yes, but solution...

$$\begin{cases}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{1}}\\{\mathrm{42x}\:+\:\mathrm{44y}\:+\:\mathrm{30z}\:=\:\mathrm{42}}\end{cases} \\ $$$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\mathrm{1},\mathrm{0},\mathrm{0}\right)\:\mathrm{yes},\:\mathrm{but}\:\mathrm{solution}... \\ $$

Question Number 209415 Answers: 1 Comments: 1

Question Number 209404 Answers: 2 Comments: 0

Question Number 209385 Answers: 1 Comments: 0

Question Number 209357 Answers: 3 Comments: 0

Evaluate : B_n = Π_(k=3) ^n (( k^( 2) −1)/(k^2 + k −6))= ?

$$ \\ $$$$\:\:\:\:\:\:{Evaluate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{B}_{{n}} =\:\underset{{k}=\mathrm{3}} {\overset{{n}} {\prod}}\:\frac{\:{k}^{\:\mathrm{2}} −\mathrm{1}}{{k}^{\mathrm{2}} \:+\:{k}\:−\mathrm{6}}=\:? \\ $$

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