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AlgebraQuestion and Answers: Page 232

Question Number 123024 Answers: 1 Comments: 0

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(√(1+(√(1+2(√(3+...(√(1+2014+(√(1+2015.2016)))))))))) = ?

$$\:\:\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{3}+...\sqrt{\mathrm{1}+\mathrm{2014}+\sqrt{\mathrm{1}+\mathrm{2015}.\mathrm{2016}}}}}}\:=\:? \\ $$

Question Number 122897 Answers: 2 Comments: 0

Question Number 122883 Answers: 4 Comments: 2

Σ_1 ^∞ (((sin (x))/x))=?

$$\sum_{\mathrm{1}} ^{\infty} \left(\frac{\mathrm{sin}\:\left(\mathrm{x}\right)}{\mathrm{x}}\right)=? \\ $$

Question Number 122860 Answers: 1 Comments: 0

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is there a formula for this? Σ_(n=0) ^∞ (1/(n^2 +an+b))

$$\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{formula}\:\mathrm{for}\:\mathrm{this}? \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{an}+{b}} \\ $$

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Question Number 122821 Answers: 1 Comments: 0

Given { ((x^(ln x+ln y) = 5)),((y^(ln x+ln y) = 2)) :} then { ((x=?)),((y=?)) :}

$$\:{Given}\:\begin{cases}{{x}^{\mathrm{ln}\:{x}+\mathrm{ln}\:{y}} \:=\:\mathrm{5}}\\{{y}^{\mathrm{ln}\:{x}+\mathrm{ln}\:{y}} \:=\:\mathrm{2}}\end{cases} \\ $$$${then}\:\begin{cases}{{x}=?}\\{{y}=?}\end{cases} \\ $$

Question Number 122658 Answers: 0 Comments: 0

we are in C^3 . (S): { ((x+y+z=2i−1 and xyz=2)),((xy+yz+xz=−2(1+i) )) :} P(z)=z^3 +(1−2i)z^2 −2(1+i)−2. show that (a,b,c) is solution of (S) if and only a;b;c are roots of P.

$${we}\:{are}\:{in}\:\mathbb{C}^{\mathrm{3}} . \\ $$$$\left({S}\right):\:\:\begin{cases}{{x}+{y}+{z}=\mathrm{2}{i}−\mathrm{1}\:{and}\:{xyz}=\mathrm{2}}\\{{xy}+{yz}+{xz}=−\mathrm{2}\left(\mathrm{1}+{i}\right)\:}\end{cases} \\ $$$${P}\left({z}\right)={z}^{\mathrm{3}} +\left(\mathrm{1}−\mathrm{2}{i}\right){z}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{1}+{i}\right)−\mathrm{2}. \\ $$$${show}\:{that}\:\left({a},{b},{c}\right)\:{is}\:{solution} \\ $$$${of}\:\left({S}\right)\:{if}\:{and}\:{only}\:{a};{b};{c}\:{are}\: \\ $$$${roots}\:{of}\:{P}. \\ $$

Question Number 122653 Answers: 1 Comments: 0

we define in base x the number y and z by: y=123^(−) ^(x ) and z=201^(−) ^x 1) without knowing x, write the product x×y×z in function of x. 2) we suppose that x+y+z=92 find x;y;z.

$${we}\:{define}\:{in}\:{base}\:{x}\:{the}\: \\ $$$${number}\:{y}\:{and}\:{z}\:{by}: \\ $$$${y}=\overline {\mathrm{123}}\:^{{x}\:} {and}\:{z}=\overline {\mathrm{201}}\:^{{x}} \\ $$$$\left.\mathrm{1}\right)\:{without}\:{knowing}\:{x},\:{write} \\ $$$${the}\:{product}\:{x}×{y}×{z}\:{in}\:{function} \\ $$$${of}\:{x}. \\ $$$$\left.\mathrm{2}\right)\:{we}\:{suppose}\:{that}\:{x}+{y}+{z}=\mathrm{92} \\ $$$${find}\:{x};{y};{z}. \\ $$$$ \\ $$

Question Number 122651 Answers: 1 Comments: 0

show that if 5^(3n) +5^(2n) +5^n +1 is divisible by 13, then n (∈ N) is not a multiple of 4.

$${show}\:{that}\:{if}\: \\ $$$$\mathrm{5}^{\mathrm{3}{n}} +\mathrm{5}^{\mathrm{2}{n}} +\mathrm{5}^{{n}} +\mathrm{1}\:{is}\:{divisible}\:{by} \\ $$$$\mathrm{13},\:{then}\:{n}\:\left(\in\:\mathbb{N}\right)\:{is}\:{not}\:{a}\: \\ $$$${multiple}\:{of}\:\mathrm{4}. \\ $$

Question Number 122649 Answers: 1 Comments: 2

By divising an integer a by integer b we find the result: 0.285714285714... followed by a group of 6 digits: 285714 which is repeated indefinited. determinate the fraction (a/b)

$${By}\:{divising}\:{an}\:{integer}\:{a}\:{by}\: \\ $$$${integer}\:{b}\:{we}\:{find}\:{the}\:{result}: \\ $$$$\mathrm{0}.\mathrm{285714285714}...\:{followed} \\ $$$${by}\:{a}\:{group}\:{of}\:\mathrm{6}\:{digits}:\:\mathrm{285714} \\ $$$${which}\:{is}\:{repeated}\:{indefinited}. \\ $$$${determinate}\:{the}\:{fraction}\:\frac{{a}}{{b}}\: \\ $$$$ \\ $$$$ \\ $$

Question Number 122648 Answers: 2 Comments: 0

Determinate the values of n ∈ N such that n−1 can divise 2n+3

$${Determinate}\:{the}\:{values}\:{of}\:{n}\:\in\:\mathbb{N} \\ $$$${such}\:{that}\:{n}−\mathrm{1}\:{can}\:{divise}\:\mathrm{2}{n}+\mathrm{3} \\ $$

Question Number 122647 Answers: 0 Comments: 0

p ∈ Z. given: u=14p+3 ; v=5p+1, (E):87x+31y=2 ; we have also the line (D): 87x−31y=2 1)show that u and v are primes between them.(i mean the don′t have any common divisor excepted 1 and −1.) 2)deduct that 87 and 31 are primes between them. 3)Solve (E). 4)Determinate points (x;y)∈ (D) which that their cordonnates x;y ∈ N and x≤100

$${p}\:\in\:\mathbb{Z}.\:{given}: \\ $$$${u}=\mathrm{14}{p}+\mathrm{3}\:;\:{v}=\mathrm{5}{p}+\mathrm{1}, \\ $$$$\left({E}\right):\mathrm{87}{x}+\mathrm{31}{y}=\mathrm{2}\:;\:{we}\:{have}\: \\ $$$${also}\:{the}\:{line}\:\left({D}\right):\:\mathrm{87}{x}−\mathrm{31}{y}=\mathrm{2} \\ $$$$\left.\mathrm{1}\right){show}\:{that}\:{u}\:{and}\:{v}\:{are}\:{primes} \\ $$$${between}\:{them}.\left({i}\:{mean}\:{the}\:\right. \\ $$$${don}'{t}\:{have}\:{any}\:{common}\:{divisor}\: \\ $$$$\left.{excepted}\:\mathrm{1}\:{and}\:−\mathrm{1}.\right) \\ $$$$\left.\mathrm{2}\right){deduct}\:{that}\:\mathrm{87}\:{and}\:\mathrm{31}\:{are} \\ $$$${primes}\:{between}\:{them}. \\ $$$$\left.\mathrm{3}\right){Solve}\:\left({E}\right). \\ $$$$\left.\mathrm{4}\right){Determinate}\:{points}\:\left({x};{y}\right)\in\:\left({D}\right)\: \\ $$$${which}\:{that}\:{their}\:{cordonnates} \\ $$$${x};{y}\:\in\:\mathbb{N}\:{and}\:{x}\leqslant\mathrm{100} \\ $$$$ \\ $$

Question Number 122646 Answers: 0 Comments: 4

Determinate the couples (a; b) such that GCD(a;b)+LCM(a;b)=a+b GCD: greatest common divisor LCM: least common multiple

$${Determinate}\:{the}\:{couples}\: \\ $$$$\left({a};\:{b}\right)\:{such}\:{that}\: \\ $$$${GCD}\left({a};{b}\right)+{LCM}\left({a};{b}\right)={a}+{b} \\ $$$${GCD}:\:{greatest}\:{common}\:{divisor} \\ $$$${LCM}:\:{least}\:{common}\:{multiple} \\ $$

Question Number 122640 Answers: 0 Comments: 0

f(f(x))=x^2 −1 find f(x)

$${f}\left({f}\left({x}\right)\right)={x}^{\mathrm{2}} −\mathrm{1} \\ $$$${find}\:{f}\left({x}\right) \\ $$

Question Number 122580 Answers: 0 Comments: 3

Solve x^x =2x

$${Solve} \\ $$$${x}^{{x}} =\mathrm{2}{x} \\ $$

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Question Number 122166 Answers: 3 Comments: 0

Given that I_n = ∫_0 ^1 x(1−x)^n dx obtain a reduction formulae for I_(n ) in terms of I_(n−2) Hence evaluate ∫_0 ^1 x(1−x)^5 dx.

$$\mathrm{Given}\:\mathrm{that}\:{I}_{{n}} \:=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}\left(\mathrm{1}−{x}\right)^{{n}} {dx} \\ $$$$\mathrm{obtain}\:\mathrm{a}\:\mathrm{reduction}\:\mathrm{formulae}\:\mathrm{for}\:{I}_{{n}\:} \:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:{I}_{{n}−\mathrm{2}} \:\mathrm{Hence}\:\mathrm{evaluate}\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}\left(\mathrm{1}−{x}\right)^{\mathrm{5}} {dx}. \\ $$

Question Number 122095 Answers: 0 Comments: 0

Find the number of triplets(x/a/b) where x is a real number and (a/b) belongs to the set {1/2/3/4/5/6/7/8/9} such that x^2 −a{x}+b =0 where {x} denotes the fractional part of real number x

$${Find}\:{the}\:\:{number}\:{of}\:{triplets}\left({x}/{a}/{b}\right)\:{where}\:{x} \\ $$$${is}\:{a}\:{real}\:{number}\:{and}\:\left({a}/{b}\right)\:{belongs}\:{to}\:{the}\:{set} \\ $$$$\left\{\mathrm{1}/\mathrm{2}/\mathrm{3}/\mathrm{4}/\mathrm{5}/\mathrm{6}/\mathrm{7}/\mathrm{8}/\mathrm{9}\right\}\:{such}\:{that}\: \\ $$$$ \\ $$$${x}^{\mathrm{2}} −{a}\left\{{x}\right\}+{b}\:=\mathrm{0} \\ $$$${where}\:\left\{{x}\right\}\:{denotes}\:{the}\:{fractional}\:{part}\:{of}\:{real}\:{number}\:{x} \\ $$$$ \\ $$$$ \\ $$

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