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Question Number 56477 Answers: 1 Comments: 0

let x and y be two real numbers if x+y≤10 prove ln(x+1)+ln(y+1)≤2ln6

$$\mathrm{let}\:{x}\:\mathrm{and}\:{y}\:\mathrm{be}\:\mathrm{two}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{if}\:{x}+{y}\leqslant\mathrm{10}\:\mathrm{prove} \\ $$$$\mathrm{ln}\left({x}+\mathrm{1}\right)+\mathrm{ln}\left({y}+\mathrm{1}\right)\leqslant\mathrm{2ln6} \\ $$

Question Number 56471 Answers: 1 Comments: 0

Question Number 56461 Answers: 1 Comments: 0

Question Number 56460 Answers: 1 Comments: 0

Question Number 56462 Answers: 0 Comments: 1

Question Number 56467 Answers: 1 Comments: 0

x^3 +ax^2 +bx+c=0 Transform to t^3 +k=0 .

$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$${Transform}\:{to}\: \\ $$$$\:\:\:{t}^{\mathrm{3}} +{k}=\mathrm{0}\:. \\ $$

Question Number 56453 Answers: 1 Comments: 2

Question Number 57028 Answers: 0 Comments: 3

Question Number 56426 Answers: 2 Comments: 0

Let consider the following sequence : 1 , 3 , (√(17)) , 5 , (√(33)) , (√(41)) , ... What may be the explicit formula that can gives this sequence of number ? Thank you

$${Let}\:{consider}\:{the}\:{following}\:{sequence}\:: \\ $$$$ \\ $$$$\mathrm{1}\:,\:\mathrm{3}\:,\:\sqrt{\mathrm{17}}\:,\:\mathrm{5}\:,\:\sqrt{\mathrm{33}}\:,\:\sqrt{\mathrm{41}}\:,\:...\: \\ $$$$ \\ $$$${What}\:{may}\:{be}\:{the}\:{explicit}\:{formula}\:{that} \\ $$$${can}\:{gives}\:{this}\:{sequence}\:{of}\:{number}\:? \\ $$$$ \\ $$$${Thank}\:{you} \\ $$

Question Number 56424 Answers: 1 Comments: 0

The sum of first 10 terms of the series (x+ (1/x))^2 + (x^2 + (1/x^2 ))^2 + (x^3 + (1/x^3 ))^2 +... is

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{10}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\left({x}+\:\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} \:+\:\left({x}^{\mathrm{2}} +\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} \:+\:\left({x}^{\mathrm{3}} +\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)^{\mathrm{2}} +...\:\mathrm{is} \\ $$

Question Number 56423 Answers: 1 Comments: 0

The sum to infinity of the series (1/3) + (1/(15)) + (1/(35)) + .... is (1/2).

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{to}\:\mathrm{infinity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{15}}\:+\:\frac{\mathrm{1}}{\mathrm{35}}\:+\:....\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{2}}. \\ $$

Question Number 56422 Answers: 1 Comments: 0

If the system of equations ax + by + (aλ+b)z = 0 bx + cy + (bλ+c)z = 0 (aλ + b)x + (bλ+c)y = 0 has a non−trivial solution, then

$$\mathrm{If}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$${ax}\:+\:{by}\:+\:\left({a}\lambda+{b}\right){z}\:=\:\mathrm{0} \\ $$$${bx}\:+\:{cy}\:+\:\left({b}\lambda+{c}\right){z}\:=\:\mathrm{0} \\ $$$$\left({a}\lambda\:+\:{b}\right){x}\:+\:\left({b}\lambda+{c}\right){y}\:=\:\mathrm{0} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{non}−\mathrm{trivial}\:\mathrm{solution},\:\mathrm{then} \\ $$

Question Number 56421 Answers: 1 Comments: 0

Let a_1 , a_2 , ..., a_(10) be in AP and h_1 , h_2 ,..., h_(10) be in HP. If a_1 = h_1 =2 and a_(10) = h_(10) =3, then a_4 h_7 is

$$\mathrm{Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} \:,\:...,\:{a}_{\mathrm{10}} \:\mathrm{be}\:\mathrm{in}\:\mathrm{AP}\:\mathrm{and}\:{h}_{\mathrm{1}} ,\:{h}_{\mathrm{2}} ,...,\:{h}_{\mathrm{10}} \\ $$$$\mathrm{be}\:\mathrm{in}\:\mathrm{HP}.\:\mathrm{If}\:\:{a}_{\mathrm{1}} =\:{h}_{\mathrm{1}} =\mathrm{2}\:\:\mathrm{and}\:\:{a}_{\mathrm{10}} =\:{h}_{\mathrm{10}} =\mathrm{3}, \\ $$$$\mathrm{then}\:{a}_{\mathrm{4}} {h}_{\mathrm{7}} \:\:\mathrm{is} \\ $$

Question Number 56420 Answers: 1 Comments: 0

Let a, b and c form a GP of common ratio r, with 0< r<1. If a, 2b and 3c form an AP, then r equals

$$\mathrm{Let}\:\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{form}\:\mathrm{a}\:\mathrm{GP}\:\mathrm{of}\:\mathrm{common}\:\mathrm{ratio}\:{r}, \\ $$$$\mathrm{with}\:\:\mathrm{0}<\:{r}<\mathrm{1}.\:\mathrm{If}\:\:{a},\:\mathrm{2}{b}\:\mathrm{and}\:\mathrm{3}{c}\:\mathrm{form}\:\mathrm{an}\:\mathrm{AP}, \\ $$$$\mathrm{then}\:{r}\:\mathrm{equals} \\ $$

Question Number 56419 Answers: 1 Comments: 0

Let a, b, c be in AP and ∣a∣<1, ∣b∣<1, ∣c∣<1. If x = 1+a+a^2 +... to ∞ y = 1+b+b^2 +... to ∞ z = 1+c+c^2 +... to ∞ then x, y, z are in

$$\mathrm{Let}\:{a},\:{b},\:{c}\:\mathrm{be}\:\mathrm{in}\:\mathrm{AP}\:\mathrm{and}\:\mid{a}\mid<\mathrm{1},\:\mid{b}\mid<\mathrm{1},\:\mid{c}\mid<\mathrm{1}.\:\mathrm{If} \\ $$$${x}\:\:=\:\:\mathrm{1}+{a}+{a}^{\mathrm{2}} +...\:\mathrm{to}\:\infty\: \\ $$$${y}\:\:=\:\:\mathrm{1}+{b}+{b}^{\mathrm{2}} +...\:\mathrm{to}\:\infty\: \\ $$$${z}\:\:=\:\:\mathrm{1}+{c}+{c}^{\mathrm{2}} +...\:\mathrm{to}\:\infty\:\: \\ $$$$\mathrm{then}\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{in} \\ $$

Question Number 56418 Answers: 1 Comments: 0

The value of a^(log_b x) where a=0.2, b=(√(5,)) x=(1/4) + (1/8) + (1/(16)) + ... to ∞ is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:{a}^{\mathrm{log}_{{b}} {x}} \mathrm{where}\:{a}=\mathrm{0}.\mathrm{2},\:{b}=\sqrt{\mathrm{5},} \\ $$$${x}=\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{1}}{\mathrm{8}}\:+\:\frac{\mathrm{1}}{\mathrm{16}}\:+\:...\:\mathrm{to}\:\infty\:\mathrm{is} \\ $$

Question Number 56417 Answers: 2 Comments: 0

The nth term of the series 4, 14, 30, 52, 80, 114, ... is

$$\mathrm{The}\:{n}\mathrm{th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series}\: \\ $$$$\mathrm{4},\:\mathrm{14},\:\mathrm{30},\:\mathrm{52},\:\mathrm{80},\:\mathrm{114},\:...\:\:\mathrm{is} \\ $$

Question Number 56416 Answers: 1 Comments: 1

(666 .... 6)^2 _(n−digits) + (888 ....8)_(n−digits) is equal to

$$\underset{{n}−\mathrm{digits}} {\left(\mathrm{666}\:....\:\mathrm{6}\right)^{\mathrm{2}} }\:+\:\underset{{n}−\mathrm{digits}} {\left(\mathrm{888}\:....\mathrm{8}\right)}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 56415 Answers: 1 Comments: 0

If for 0<x<(π/2) , y=exp[(sin^2 x+sin^4 x+sin^6 x+...∞)log_e 2] is a zero of the quadratic equation x^2 −9x+8=0, then the value of((sin x+cos x)/(sin x−cos x)) is

$$\mathrm{If}\:\:\:\mathrm{for}\:\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}}\:, \\ $$$$\:{y}={exp}\left[\left(\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{sin}^{\mathrm{4}} {x}+\mathrm{sin}^{\mathrm{6}} {x}+...\infty\right)\mathrm{log}_{{e}} \mathrm{2}\right] \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{zero}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$${x}^{\mathrm{2}} −\mathrm{9}{x}+\mathrm{8}=\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}\:\mathrm{is} \\ $$

Question Number 56414 Answers: 0 Comments: 0

If for 0<x<(π/2) , y=exp[(sin^2 x+sin^4 x+sin^6 x+...∞)log_e 2] is a zero of the quadratic equation x^2 −9x+8=0, then the value of((sin x+cos x)/(sin x−cos x)) is

Question Number 56413 Answers: 0 Comments: 1

The solution of the equation 1+a+a^2 +a^3 +...+a^x =(1+a)(1+a^2 )(1+a^4 ) is given by x = _____.

$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{1}+{a}+{a}^{\mathrm{2}} +{a}^{\mathrm{3}} +...+{a}^{{x}} =\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{a}^{\mathrm{4}} \right) \\ $$$$\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:{x}\:=\:\_\_\_\_\_. \\ $$

Question Number 56411 Answers: 1 Comments: 0

For a sequence <a_n > ; a_1 =2, (a_(n+1) /a_n ) = (1/3), then Σ_(r=1) ^(20) a_r is equal to

$$\mathrm{For}\:\mathrm{a}\:\mathrm{sequence}\:<{a}_{{n}} >\:;\:{a}_{\mathrm{1}} =\mathrm{2},\:\frac{{a}_{{n}+\mathrm{1}} }{{a}_{{n}} }\:=\:\frac{\mathrm{1}}{\mathrm{3}}, \\ $$$$\mathrm{then}\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\:{a}_{{r}} \:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 56404 Answers: 1 Comments: 0

Question Number 56392 Answers: 2 Comments: 2

find the integral of ∫x(√(x−1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{of}\:\:\int\mathrm{x}\sqrt{\mathrm{x}−\mathrm{1}}\mathrm{dx} \\ $$

Question Number 56401 Answers: 2 Comments: 1

∫((x+sin x)/(1+cos x))dx

$$\int\frac{{x}+\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{cos}\:{x}}{dx} \\ $$

Question Number 56390 Answers: 1 Comments: 2

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