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Question Number 142595 Answers: 1 Comments: 0

Prove :: sec^2 x=4Σ_(n=0) ^∞ {(1/([(2n+1)π−2x]^2 ))+(1/([(2n+1)π+2x]^2 ))}

$$\mathrm{Prove}\:::\:\:\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}=\mathrm{4}\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\mathrm{1}}{\left[\left(\mathrm{2n}+\mathrm{1}\right)\pi−\mathrm{2x}\right]^{\mathrm{2}} }+\frac{\mathrm{1}}{\left[\left(\mathrm{2n}+\mathrm{1}\right)\pi+\mathrm{2x}\right]^{\mathrm{2}} }\right\} \\ $$

Question Number 142618 Answers: 0 Comments: 2

In how many ways can committee of 5 be formed from a group of 11 people consisting of 4 teachers and 7 students if there is no restriction in the selection ? _______________________

$$\:\:\:{In}\:{how}\:{many}\:{ways}\:{can}\:{committee} \\ $$$${of}\:\mathrm{5}\:{be}\:{formed}\:{from}\:{a}\:{group}\: \\ $$$${of}\:\mathrm{11}\:{people}\:{consisting}\:{of}\:\mathrm{4}\:{teachers} \\ $$$${and}\:\mathrm{7}\:{students}\:{if}\:{there}\:{is}\:{no}\: \\ $$$${restriction}\:{in}\:{the}\:{selection}\:? \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$

Question Number 142617 Answers: 0 Comments: 0

Question Number 142613 Answers: 0 Comments: 0

Evaluate ∫_0 ^1 ((log(x)log((x/(1−x))))/( (√(x/(1−x)))))dx

$$\boldsymbol{\mathrm{Evaluate}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{log}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}}\right)}{\:\sqrt{\frac{\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}}}}\boldsymbol{\mathrm{dx}} \\ $$$$ \\ $$

Question Number 142592 Answers: 1 Comments: 0

Is there any android apk compute generating function GF = Π_(i=1) ^(m) (Σ_(k=1) ^(n_i ) C_k ^( n_i ) x^k ) thank you so much

$${Is}\:{there}\:{any}\:{android}\:{apk} \\ $$$${compute}\:{generating}\:{function} \\ $$$${GF}\:=\:\underset{{i}=\mathrm{1}} {\overset{{m}} {\Pi}}\:\left(\underset{{k}=\mathrm{1}} {\overset{{n}_{{i}} } {\Sigma}}{C}_{{k}} ^{\:{n}_{{i}} } \:{x}^{{k}} \right) \\ $$$${thank}\:{you}\:{so}\:{much} \\ $$

Question Number 142573 Answers: 2 Comments: 0

Question Number 142865 Answers: 1 Comments: 1

Let a_1 , a_2 , a_3 , ... be an arithmethic progression of positive real numbers. Then (1/( (√a_1 )+(√a_2 )))+(1/( (√a_2 )+(√a_3 )))+∙∙∙+(1/( (√a_(n−1) )+(√a_n )))= (A) ((n+1)/( (√a_1 )+(√a_n ))) (B) ((n−1)/( (√a_1 )+(√a_n ))) (C) (n/( (√a_1 )+(√a_n ))) (D) (n/( (√a_n )−(√a_1 )))

$$\mathrm{Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:...\:\mathrm{be}\:\mathrm{an}\:\mathrm{arithmethic}\:\mathrm{progression}\:\mathrm{of} \\ $$$$\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}.\:\mathrm{Then} \\ $$$$\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{{a}_{\mathrm{1}} }+\sqrt{{a}_{\mathrm{2}} }}+\frac{\mathrm{1}}{\:\sqrt{{a}_{\mathrm{2}} }+\sqrt{{a}_{\mathrm{3}} }}+\centerdot\centerdot\centerdot+\frac{\mathrm{1}}{\:\sqrt{{a}_{{n}−\mathrm{1}} }+\sqrt{{a}_{{n}} }}= \\ $$$$\left(\mathrm{A}\right)\:\frac{{n}+\mathrm{1}}{\:\sqrt{{a}_{\mathrm{1}} }+\sqrt{{a}_{{n}} }}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\frac{{n}−\mathrm{1}}{\:\sqrt{{a}_{\mathrm{1}} }+\sqrt{{a}_{{n}} }} \\ $$$$\left(\mathrm{C}\right)\:\frac{{n}}{\:\sqrt{{a}_{\mathrm{1}} }+\sqrt{{a}_{{n}} }}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\frac{{n}}{\:\sqrt{{a}_{{n}} }−\sqrt{{a}_{\mathrm{1}} }} \\ $$

Question Number 142570 Answers: 1 Comments: 0

Find all functions f:R→R such that f(x+y)=2f(x)+3f(y)−4xyf(2x−3y) (∀x;y∈R)

$${Find}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{such}\:{that} \\ $$$${f}\left({x}+{y}\right)=\mathrm{2}{f}\left({x}\right)+\mathrm{3}{f}\left({y}\right)−\mathrm{4}{xyf}\left(\mathrm{2}{x}−\mathrm{3}{y}\right) \\ $$$$\left(\forall{x};{y}\in\mathbb{R}\right) \\ $$

Question Number 142582 Answers: 2 Comments: 0

z^4 + 12z + 3 = 0 (z=?)

$$\boldsymbol{{z}}^{\mathrm{4}} \:+\:\mathrm{12}\boldsymbol{{z}}\:+\:\mathrm{3}\:=\:\mathrm{0}\:\left(\boldsymbol{{z}}=?\right) \\ $$

Question Number 142577 Answers: 2 Comments: 0

Let a,b,c ≥ 0 and a+b+c = 1.Prove that (1/(a^2 +1))+(1/(b^2 +1))+(1/(c^2 +1)) ≥ (5/2)

$$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{1}.\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{1}}{{b}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{1}}{{c}^{\mathrm{2}} +\mathrm{1}}\:\geqslant\:\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 142560 Answers: 1 Comments: 0