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AlgebraQuestion and Answers: Page 250

Question Number 98205    Answers: 1   Comments: 0

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

$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{nth}}\:\:\boldsymbol{\mathrm{term}}\:\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:\:=\:\:\mathrm{1},\:\:\:\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}} \:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:\:+\:\:\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:−\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\:−\:\:\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}\right)^{\mathrm{2}} }\:\:\:\:\left(\boldsymbol{\mathrm{n}}\:\:=\:\:\mathrm{1},\:\:\mathrm{2},\:\:\mathrm{3},\:\:...\right) \\ $$

Question Number 98192    Answers: 2   Comments: 1

Question Number 98293    Answers: 2   Comments: 0

Question Number 98130    Answers: 0   Comments: 0

Question Number 98077    Answers: 0   Comments: 4

Question Number 98073    Answers: 1   Comments: 2

solve (√(6−x)) = 6−x^2

$$\mathrm{solve}\:\sqrt{\mathrm{6}−\mathrm{x}}\:=\:\mathrm{6}−\mathrm{x}^{\mathrm{2}} \\ $$

Question Number 98067    Answers: 2   Comments: 0

Question Number 97968    Answers: 1   Comments: 1

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Question Number 99300    Answers: 1   Comments: 0

Find Σ_(n=1) ^∞ (1/((3n)!))=?

$$\mathrm{Find}\:\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}\boldsymbol{{n}}\right)!}=? \\ $$

Question Number 97936    Answers: 1   Comments: 1

Find the value of (√(45−(√(2000)) )) + (√(45+(√(2000))))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\sqrt{\mathrm{45}−\sqrt{\mathrm{2000}}\:}\:\:+\:\:\sqrt{\mathrm{45}+\sqrt{\mathrm{2000}}}\: \\ $$

Question Number 97920    Answers: 0   Comments: 4

Question Number 97919    Answers: 1   Comments: 2

Question Number 97818    Answers: 1   Comments: 0

if y^2 = ax^2 + bx + c Show that: y (d^3 y/dx^3 ) + 3 (dy/dx) (d^2 y/dx^2 ) = 0

$$\boldsymbol{\mathrm{if}}\:\:\:\:\:\boldsymbol{\mathrm{y}}^{\mathrm{2}} \:\:=\:\:\boldsymbol{\mathrm{ax}}^{\mathrm{2}} \:+\:\boldsymbol{\mathrm{bx}}\:+\:\:\boldsymbol{\mathrm{c}} \\ $$$$\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{3}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{3}} }\:\:+\:\:\mathrm{3}\:\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}} }\:\:\:=\:\:\:\mathrm{0} \\ $$

Question Number 97808    Answers: 0   Comments: 1

Question Number 97694    Answers: 1   Comments: 7

Question Number 97675    Answers: 1   Comments: 0

Question Number 97637    Answers: 0   Comments: 1

Given p,q∈R_+ ^∗ −{−1}/(1/p)+(1/q)=1 show that; ∀a,b ∈R ab≤(a^p /p)+(b^q /q)

$$\mathrm{Given}\:\mathrm{p},\mathrm{q}\in\mathbb{R}_{+} ^{\ast} −\left\{−\mathrm{1}\right\}/\frac{\mathrm{1}}{\mathrm{p}}+\frac{\mathrm{1}}{\mathrm{q}}=\mathrm{1}\:\mathrm{show}\:\mathrm{that}; \\ $$$$\forall\mathrm{a},\mathrm{b}\:\in\mathbb{R}\:\mathrm{ab}\leqslant\frac{\mathrm{a}^{\mathrm{p}} }{\mathrm{p}}+\frac{\mathrm{b}^{\mathrm{q}} }{\mathrm{q}} \\ $$

Question Number 97564    Answers: 1   Comments: 0

Question Number 97512    Answers: 0   Comments: 2

The value of k for which the quadratic equation (1−2k)x^2 −6kx−1=0 and kx^2 −x+1=0 have atleast one roots in common are ___

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the} \\ $$$$\mathrm{quadratic}\:\mathrm{equation}\:\left(\mathrm{1}−\mathrm{2k}\right)\mathrm{x}^{\mathrm{2}} −\mathrm{6kx}−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{kx}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}=\mathrm{0}\:\mathrm{have}\:\mathrm{atleast} \\ $$$$\mathrm{one}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{common}\:\mathrm{are}\:\_\_\_ \\ $$

Question Number 97501    Answers: 0   Comments: 2

The natural number n for which the expression y = 5log^2 _3 (n) − log _3 (n^(12) )+9 , has the minimum value is ___

$$\mathrm{The}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\mathrm{the}\:\mathrm{expression}\:\mathrm{y}\:=\:\mathrm{5log}^{\mathrm{2}} \:_{\mathrm{3}} \left(\mathrm{n}\right)\:− \\ $$$$\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{n}^{\mathrm{12}} \right)+\mathrm{9}\:,\:\mathrm{has}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{is}\:\_\_\_ \\ $$

Question Number 97496    Answers: 1   Comments: 0

if f(((2x+5)/(x−3)))=3x+5 find f(x) please solve it

$${if}\:\:\:\:\:\:{f}\left(\frac{\mathrm{2}{x}+\mathrm{5}}{{x}−\mathrm{3}}\right)=\mathrm{3}{x}+\mathrm{5}\:\:\:{find}\:\:\:{f}\left({x}\right) \\ $$$$ \\ $$$${please}\:{solve}\:{it} \\ $$

Question Number 97485    Answers: 1   Comments: 3

Question Number 97428    Answers: 2   Comments: 5

Question Number 97386    Answers: 0   Comments: 2

Question Number 97306    Answers: 0   Comments: 2

if sin14=x then cos^2 22−cos^2 8=?

$$\mathrm{if}\:\:\:\:\:\:\:\:\:\mathrm{sin14}=\mathrm{x} \\ $$$$\mathrm{then} \\ $$$$\mathrm{cos}^{\mathrm{2}} \mathrm{22}−\mathrm{cos}^{\mathrm{2}} \mathrm{8}=? \\ $$

Question Number 97303    Answers: 0   Comments: 0

Given x_1 +x_2 +x_3 = 0 , y_1 + y_2 +y_3 = 0 and x_1 y_1 + x_2 y_2 + x_3 y_3 = 0 . The value of (x_1 ^2 /(x_1 ^2 +x_2 ^2 +x_3 ^2 )) + (y_1 ^2 /(y_1 ^2 +y_2 ^2 +y_3 ^2 )) = ?

$$\boldsymbol{\mathrm{G}}\mathrm{iven}\:\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} \:=\:\mathrm{0}\:,\:\mathrm{y}_{\mathrm{1}} \:+\:\mathrm{y}_{\mathrm{2}} +\mathrm{y}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{x}_{\mathrm{1}} \mathrm{y}_{\mathrm{1}} +\:\mathrm{x}_{\mathrm{2}} \mathrm{y}_{\mathrm{2}} \:+\:\mathrm{x}_{\mathrm{3}} \mathrm{y}_{\mathrm{3}} \:=\:\mathrm{0}\:.\:\mathrm{The}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} ^{\mathrm{2}} }\:+\:\frac{\mathrm{y}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{y}_{\mathrm{1}} ^{\mathrm{2}} \:+\mathrm{y}_{\mathrm{2}} ^{\mathrm{2}} \:+\mathrm{y}_{\mathrm{3}} ^{\mathrm{2}} }\:=\:?\: \\ $$

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