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AlgebraQuestion and Answers: Page 310

Question Number 61137 Answers: 1 Comments: 0

What is the sum of first 3n term of an AP , if the sunm of first n term is 2n and sum of first 2n term is 5n

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{3n}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:,\:\mathrm{if}\:\mathrm{the}\:\mathrm{sunm}\:\mathrm{of}\:\mathrm{first}\:\mathrm{n}\:\mathrm{term}\:\mathrm{is} \\ $$$$\mathrm{2n}\:\:\mathrm{and}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{2n}\:\mathrm{term}\:\mathrm{is}\:\:\mathrm{5n} \\ $$

Question Number 61117 Answers: 2 Comments: 0

The 2nd, 4th and 8th term of an AP are the consecutive term of a GP. If the sum of the 3rd and 4th term of the AP is 20. Find the sum of the first four terms of the AP.

$$\mathrm{The}\:\mathrm{2nd},\:\mathrm{4th}\:\mathrm{and}\:\mathrm{8th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{are}\:\mathrm{the}\:\mathrm{consecutive}\:\mathrm{term}\:\mathrm{of}\:\mathrm{a}\:\mathrm{GP}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{3rd}\:\mathrm{and}\:\mathrm{4th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{AP}\:\mathrm{is}\:\mathrm{20}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{four}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{AP}. \\ $$

Question Number 61111 Answers: 1 Comments: 2

Please what does the 2 on the C mean. C_1 ^2 + 2 C_2 ^2 + 3 C_3 ^2 + ... + n C_n ^2 = (((2n − 1)!)/([(n − 1)!]^2 )) Does the 2 on C mean square ?? I mean: (C_1 )^2 + 2(C_2 )^2 + 3(C_3 )^2 + ... + n (C_n )^2 which is also ( ^n C_1 )^2 + 2( ^n C_2 )^2 + 3( ^n C_3 )^2 + ... + n ( ^n C_n )^2 I just want to know what the 2 on C represent . Thanks. C_1 ^2 + 2 C_2 ^2 + 3 C_3 ^2 + ... + n C_n ^2 = (((2n − 1)!)/([(n − 1)!]^2 ))

Question Number 60980 Answers: 1 Comments: 2

Solve for x, y, z x(y + z) = 33 ..... (i) y(z + x) = 35 ..... (ii) z(x + y) = 14 ..... (iii)

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x},\:\mathrm{y},\:\mathrm{z} \\ $$$$\:\:\:\:\:\mathrm{x}\left(\mathrm{y}\:+\:\mathrm{z}\right)\:=\:\mathrm{33}\:\:\:\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\mathrm{y}\left(\mathrm{z}\:+\:\mathrm{x}\right)\:=\:\mathrm{35}\:\:\:\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\:\mathrm{z}\left(\mathrm{x}\:+\:\mathrm{y}\right)\:=\:\mathrm{14}\:\:\:\:\:\:\:\:.....\:\left(\mathrm{iii}\right) \\ $$

Question Number 60946 Answers: 3 Comments: 4

Find x: x^x = 2x

$$\mathrm{Find}\:\mathrm{x}:\:\:\:\:\:\:\mathrm{x}^{\mathrm{x}} \:\:=\:\:\mathrm{2x} \\ $$

Question Number 60905 Answers: 3 Comments: 0

if x+(1/x)=(√3).find x^(24) +x^(18) +x^6 +1

$${if}\:{x}+\frac{\mathrm{1}}{{x}}=\sqrt{\mathrm{3}}.{find} \\ $$$${x}^{\mathrm{24}} +{x}^{\mathrm{18}} +{x}^{\mathrm{6}} +\mathrm{1} \\ $$

Question Number 60910 Answers: 1 Comments: 7

Question Number 60854 Answers: 2 Comments: 4

(x^4 −3x^2 +2x+1)/(x−1)

$$\left({x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right)/\left({x}−\mathrm{1}\right) \\ $$$$ \\ $$$$ \\ $$

Question Number 60817 Answers: 4 Comments: 2

V=(4/3)𝛑R^3 prove

$$\boldsymbol{\mathrm{V}}=\frac{\mathrm{4}}{\mathrm{3}}\boldsymbol{\pi\mathrm{R}}^{\mathrm{3}} \:\:\:\boldsymbol{\mathrm{prove}} \\ $$

Question Number 60816 Answers: 1 Comments: 0

S=4𝛑R^2 prove

$$\boldsymbol{\mathrm{S}}=\mathrm{4}\boldsymbol{\pi\mathrm{R}}^{\mathrm{2}} \:\:\:\boldsymbol{\mathrm{prove}} \\ $$

Question Number 60745 Answers: 4 Comments: 5

Question Number 60734 Answers: 1 Comments: 4

Find the product of the real roots of the equation (x + 2 + (√(x^2 + 4x + 3)))^5 − 32(x + 2 − (√(x^2 + 4x + 3)))^5 = 31

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{x}\:+\:\mathrm{2}\:+\:\sqrt{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4x}\:+\:\mathrm{3}}\right)^{\mathrm{5}} \:−\:\:\mathrm{32}\left(\mathrm{x}\:+\:\mathrm{2}\:−\:\sqrt{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4x}\:+\:\mathrm{3}}\right)^{\mathrm{5}} \:\:=\:\:\mathrm{31} \\ $$

Question Number 60723 Answers: 0 Comments: 10

solve for x (√(a−(√(a+x)))) + (√(a+(√(a−x)))) = 2x

$${solve}\:{for}\:{x}\: \\ $$$$\sqrt{{a}−\sqrt{{a}+{x}}}\:+\:\sqrt{{a}+\sqrt{{a}−{x}}}\:=\:\mathrm{2}{x} \\ $$

Question Number 60705 Answers: 1 Comments: 0

Question Number 60533 Answers: 1 Comments: 2

If A, B, C are angle of a triangle. Show that cos (1/2)C + cos (1/2)(A − B) = 2 sin (1/2)A sin (1/2)B

$$\mathrm{If}\:\:\mathrm{A},\:\mathrm{B},\:\mathrm{C}\:\:\mathrm{are}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}.\:\mathrm{Show}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{C}\:+\:\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{A}\:−\:\mathrm{B}\right)\:\:=\:\:\mathrm{2}\:\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{A}\:\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{B} \\ $$

Question Number 60501 Answers: 1 Comments: 1

let A = ((( 1 1)),((1 1)) ) 1)calculate A^n 2) determine e^A and e^(−A) .

$${let}\:{A}\:=\begin{pmatrix}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{e}^{{A}} \:\:\:{and}\:{e}^{−{A}} \:. \\ $$$$ \\ $$

Question Number 60500 Answers: 0 Comments: 2

let A = (((1 1)),((−2 3)) ) 1) find A^(−1) 2) calculate A^n 3) determine e^A and e^(−2A) .

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\\{−\mathrm{2}\:\:\:\mathrm{3}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}^{−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{3}\right)\:{determine}\:{e}^{{A}} \:\:\:{and}\:{e}^{−\mathrm{2}{A}} \:. \\ $$

Question Number 60475 Answers: 1 Comments: 0

Question Number 60453 Answers: 0 Comments: 0

Question Number 60445 Answers: 2 Comments: 1

Question Number 60416 Answers: 0 Comments: 0

Sum the series: ^n C_0 ^n C_1 + ^n C_1 ^n C_2 + ^n C_2 ^n C_3 + ... + ^n C_r ^n C_(r + 1)

$$\mathrm{Sum}\:\mathrm{the}\:\mathrm{series}:\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{0}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{1}} \:+\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{1}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{2}} \:+\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{2}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{3}} \:+\:...\:+\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}\:+\:\mathrm{1}} \\ $$

Question Number 60406 Answers: 1 Comments: 0

n ∈ Z^+ , Find the coefficient of x^(−1) in the expansion of (1 + x)^n (1 + (1/x))^n

$$\mathrm{n}\:\in\:\mathbb{Z}^{+} ,\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\:\mathrm{x}^{−\mathrm{1}} \:\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\:\left(\mathrm{1}\:+\:\mathrm{x}\right)^{\mathrm{n}} \left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{n}} \\ $$

Question Number 60330 Answers: 0 Comments: 1

Question Number 60255 Answers: 0 Comments: 0

b=(((kT)/P))^(1/3) . distance molekular prove.

$$\boldsymbol{\mathrm{b}}=\sqrt[{\mathrm{3}}]{\frac{\boldsymbol{\mathrm{kT}}}{\boldsymbol{\mathrm{P}}}}.\:\boldsymbol{\mathrm{distance}}\:\:\boldsymbol{\mathrm{molekular}} \\ $$$$\boldsymbol{\mathrm{prove}}. \\ $$

Question Number 60175 Answers: 0 Comments: 2

solving u^v =w with u, v, w ∈C finding all possible solutions I tested this with several values and found no mistake. please review and comment. I hope this will help at least some of you.

$$\mathrm{solving}\:{u}^{{v}} ={w}\:\mathrm{with}\:{u},\:{v},\:{w}\:\in\mathbb{C} \\ $$$$\mathrm{finding}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{solutions} \\ $$$$\mathrm{I}\:\mathrm{tested}\:\mathrm{this}\:\mathrm{with}\:\mathrm{several}\:\mathrm{values}\:\mathrm{and}\:\mathrm{found} \\ $$$$\mathrm{no}\:\mathrm{mistake}.\:\mathrm{please}\:\mathrm{review}\:\mathrm{and}\:\mathrm{comment}. \\ $$$$\mathrm{I}\:\mathrm{hope}\:\mathrm{this}\:\mathrm{will}\:\mathrm{help}\:\mathrm{at}\:\mathrm{least}\:\mathrm{some}\:\mathrm{of}\:\mathrm{you}. \\ $$

Question Number 60156 Answers: 3 Comments: 2

Prove by principle of mathematical induction sin(x) + sin(2x) + sin(3x) + ... + sin(nx) = ((cos((1/2)x) − cos(n + (1/2))x)/(2 sin((1/2)x)))

$$\mathrm{Prove}\:\mathrm{by}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{sin}\left(\mathrm{x}\right)\:+\:\mathrm{sin}\left(\mathrm{2x}\right)\:+\:\mathrm{sin}\left(\mathrm{3x}\right)\:+\:...\:+\:\mathrm{sin}\left(\mathrm{nx}\right)\:\:=\:\:\frac{\mathrm{cos}\left(\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}\right)\:−\:\mathrm{cos}\left(\mathrm{n}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right)\mathrm{x}}{\mathrm{2}\:\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}\right)} \\ $$

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