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Question Number 113336 Answers: 0 Comments: 0

If 1, a^2 ,a^3 ,...,a^(n−1) are the roots nth of unity , prove that : (1+a)(1+a^2 )(1+a^3 )...(1+a^(n−1) ) = n−2⌊(n/2)⌋

$${If}\:\mathrm{1},\:{a}^{\mathrm{2}} ,{a}^{\mathrm{3}} \:,...,{a}^{{n}−\mathrm{1}} \:{are}\:{the}\:{roots}\: \\ $$$${nth}\:{of}\:{unity}\:,\: \\ $$$${prove}\:{that}\::\:\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{a}^{\mathrm{3}} \right)...\left(\mathrm{1}+{a}^{{n}−\mathrm{1}} \right) \\ $$$$=\:{n}−\mathrm{2}\lfloor\frac{{n}}{\mathrm{2}}\rfloor \\ $$$$ \\ $$

Question Number 113190 Answers: 1 Comments: 3

∫_0 ^1 ((logx)/(x−1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{logx}}{{x}−\mathrm{1}}{dx} \\ $$

Question Number 113185 Answers: 1 Comments: 0

Question Number 112838 Answers: 1 Comments: 0

What is the sum of all the solutions of the equation ∣2x+8∣^2 −∣9x+36∣−9=0

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mid\mathrm{2x}+\mathrm{8}\mid^{\mathrm{2}} −\mid\mathrm{9x}+\mathrm{36}\mid−\mathrm{9}=\mathrm{0} \\ $$

Question Number 112809 Answers: 1 Comments: 0

Question Number 112707 Answers: 0 Comments: 0

Question Number 112454 Answers: 1 Comments: 1

(1) find the locus ∣z−z_1 ∣ = 2 meets the positive real axis (2)On a single Argand diagram, sketch the loci → { ((∣z−z_1 ∣=2)),((arg(z−z_2 )=(π/4))) :}

$$\left(\mathrm{1}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{locus}\:\mid\mathrm{z}−\mathrm{z}_{\mathrm{1}} \mid\:=\:\mathrm{2}\:\mathrm{meets} \\ $$$$\mathrm{the}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{axis} \\ $$$$\left(\mathrm{2}\right)\mathrm{On}\:\mathrm{a}\:\mathrm{single}\:\mathrm{Argand}\:\mathrm{diagram},\:\mathrm{sketch} \\ $$$$\mathrm{the}\:\mathrm{loci}\:\rightarrow\begin{cases}{\mid\mathrm{z}−\mathrm{z}_{\mathrm{1}} \mid=\mathrm{2}}\\{\mathrm{arg}\left(\mathrm{z}−\mathrm{z}_{\mathrm{2}} \right)=\frac{\pi}{\mathrm{4}}}\end{cases} \\ $$

Question Number 112430 Answers: 0 Comments: 0

Π_(n=1) ^∞ (1/(1−x^n ))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\frac{\mathrm{1}}{\mathrm{1}−{x}^{{n}} } \\ $$

Question Number 112245 Answers: 0 Comments: 0

(1/(1+x^2 ))+(1/(1+x^4 ))+(1/(1+x^8 ))+..... ( x>1)

$$\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{4}} }+\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{8}} }+.....\:\:\:\:\:\:\:\:\:\:\left(\:{x}>\mathrm{1}\right) \\ $$

Question Number 112162 Answers: 1 Comments: 3

Question Number 111975 Answers: 1 Comments: 0

Question Number 111957 Answers: 0 Comments: 31

“MATHEMATICS” CONTAINS ALL THE LETTERS OF “ETHICS”. IS THERE ANY LESSON FOR US IN ABOVE SAYING? FOR “math-lovers”? FOR “math-giants”? FOR “overflow-mathematicians”? ........ ...... _( BTW this saying belongs to me)

Question Number 111774 Answers: 1 Comments: 0

Question Number 111668 Answers: 0 Comments: 6

Question Number 111643 Answers: 1 Comments: 0

((7/3))!(with out calculator)

$$\left(\frac{\mathrm{7}}{\mathrm{3}}\right)!\left({with}\:{out}\:{calculator}\right) \\ $$

Question Number 111536 Answers: 2 Comments: 2

Let 2,3,5,6,7,10,11,... be increasing sequence of positive integers that are neither the square nor cube of an integer. Find the 2016th term of this sequence.

$$\mathrm{Let}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{10},\mathrm{11},...\:\mathrm{be}\:\mathrm{increasing} \\ $$$$\mathrm{sequence}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{neither}\:\mathrm{the}\:\mathrm{square}\:\mathrm{nor}\:\mathrm{cube}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{integer}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{2016th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{sequence}. \\ $$

Question Number 111466 Answers: 1 Comments: 2

Σ_(n=1) ^∞ (n^n /(n!))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{{n}} }{{n}!} \\ $$

Question Number 111450 Answers: 0 Comments: 1