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Question Number 175821 Answers: 1 Comments: 1

Question Number 175815 Answers: 2 Comments: 0

If n≥1 a and b are positive real numbers Then prove that: ((a^n + b^n )/(a + b)) ≥ ((a^(n−1) + b^(n−1) )/2)

$$\mathrm{If}\:\:\:\mathrm{n}\geqslant\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{a}^{\boldsymbol{\mathrm{n}}} \:\:+\:\:\mathrm{b}^{\boldsymbol{\mathrm{n}}} }{\mathrm{a}\:\:+\:\:\mathrm{b}}\:\:\geqslant\:\:\frac{\mathrm{a}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \:\:+\:\:\mathrm{b}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} }{\mathrm{2}} \\ $$

Question Number 175814 Answers: 2 Comments: 1

Question Number 175806 Answers: 1 Comments: 0

Question Number 175801 Answers: 2 Comments: 5

solve the follwing equation x(√(x )) + y(√(y )) = 3 and x(√(y )) + y(√(x )) = 2 someone solve the above equations in the following way x^3 + y^3 + 2xy(√(xy )) = 9.....(1) and x^2 y + y^2 x + 2xy(√(xy )) = 4......(2) (1) − (2) ⇒ (x − y)(x^2 − y^2 ) = 5 hence x = 3 and y = 2 which is obiviusly does not satisfy the original equations. where is the fallacy in the above solution? Please explain.

$${solve}\:{the}\:{follwing}\:{equation} \\ $$$${x}\sqrt{{x}\:}\:\:\:+\:\:{y}\sqrt{{y}\:}\:\:=\:\:\mathrm{3}\:\:\:{and}\:\:\:{x}\sqrt{{y}\:}\:\:+\:{y}\sqrt{{x}\:}\:\:=\:\:\mathrm{2} \\ $$$${someone}\:{solve}\:{the}\:{above}\:{equations}\:{in}\:{the}\:{following}\:{way}\: \\ $$$${x}^{\mathrm{3}} +\:{y}^{\mathrm{3}} +\:\mathrm{2}{xy}\sqrt{{xy}\:}\:\:=\:\mathrm{9}.....\left(\mathrm{1}\right)\:\:\:\:{and}\:\:\:{x}^{\mathrm{2}} {y}\:\:+\:\:{y}^{\mathrm{2}} {x}\:\:+\:\mathrm{2}{xy}\sqrt{{xy}\:}\:\:=\:\:\mathrm{4}......\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{1}\right)\:−\:\left(\mathrm{2}\right)\:\:\:\Rightarrow\:\:\left({x}\:−\:{y}\right)\left({x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \:\right)\:=\:\mathrm{5} \\ $$$${hence}\:\:{x}\:=\:\mathrm{3}\:\:{and}\:\:{y}\:=\:\mathrm{2}\:\:{which}\:{is}\:{obiviusly}\:{does}\:{not}\:\:{satisfy}\:{the} \\ $$$$\:{original}\:{equations}. \\ $$$${where}\:{is}\:{the}\:{fallacy}\:\:{in}\:{the}\:{above}\:{solution}?\:\:\mathrm{Please}\:\mathrm{explain}. \\ $$

Question Number 175800 Answers: 1 Comments: 0

Find prime numbers of 3 digits such that equal to sum of 3 diffrent numbers of prime

$$\:\mathrm{Find}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{of} \\ $$$$\:\:\mathrm{3}\:\mathrm{digits}\:\mathrm{such}\:\mathrm{that}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\:\mathrm{sum}\:\mathrm{of}\:\mathrm{3}\:\mathrm{diffrent}\:\mathrm{numbers}\:\mathrm{of} \\ $$$$\:\:\mathrm{prime} \\ $$

Question Number 175793 Answers: 0 Comments: 2

if xy+y^2 +zx = 48; where x,y,z are three positive real numbers then find the maximum possible value of the product (xyz)

$$\:\:\mathrm{if}\:\mathrm{xy}+\mathrm{y}^{\mathrm{2}} +\mathrm{zx}\:=\:\mathrm{48};\:\mathrm{where}\:\mathrm{x},\mathrm{y},\mathrm{z} \\ $$$$\:\:\mathrm{are}\:\mathrm{three}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\:\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible} \\ $$$$\:\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{product}\:\left(\mathrm{xyz}\right) \\ $$

Question Number 175799 Answers: 2 Comments: 0

For x ,y ε Z^+ such that 7x+9y=405. Find max value of x−y.

$$\:\:\mathrm{For}\:\mathrm{x}\:,\mathrm{y}\:\varepsilon\:\mathbb{Z}^{+} \:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\mathrm{7x}+\mathrm{9y}=\mathrm{405}.\:\mathrm{Find}\:\mathrm{max}\:\mathrm{value} \\ $$$$\:\:\mathrm{of}\:\mathrm{x}−\mathrm{y}. \\ $$

Question Number 175785 Answers: 2 Comments: 3

Question Number 175780 Answers: 0 Comments: 0

find the range of the function f(x) = cosx{sinx + (√(sin^2 x + sin^2 α )) }

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{cosx}\left\{\mathrm{sin}{x}\:+\:\sqrt{\mathrm{sin}^{\mathrm{2}} {x}\:+\:\mathrm{sin}^{\mathrm{2}} \alpha\:\:}\:\right\}\:\: \\ $$

Question Number 175770 Answers: 0 Comments: 0

P_n = e^( ((1/1) −(1/2)) +((1/3) −(1/4)) +...+((1/(n−1)) −(1/n))) = e^( (1−(1/2) +(1/3) −(1/4) +...+(1/(n−1)) −(1/n))) = e^( Σ_(k=1) ^n (( (−1 )^( k+1) )/k)) ∴ P = lim_( n→∞) (e^( Σ_(k=1) ^n (((−1)^( k+1) )/k)) ) = e^( lim_( n→∞) ( Σ_(k=1) ^n (((−1)^(k+1) )/k))) = e^( ln(2)) = 2

$$ \\ $$$$\:\:\:\mathrm{P}_{{n}} \:=\:{e}^{\:\left(\frac{\mathrm{1}}{\mathrm{1}}\:−\frac{\mathrm{1}}{\mathrm{2}}\right)\:+\left(\frac{\mathrm{1}}{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{4}}\right)\:+...+\left(\frac{\mathrm{1}}{{n}−\mathrm{1}}\:−\frac{\mathrm{1}}{{n}}\right)} \\ $$$$\:\:\:\:\:\:\:\:=\:{e}^{\:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{4}}\:\:+...+\frac{\mathrm{1}}{{n}−\mathrm{1}}\:−\frac{\mathrm{1}}{{n}}\right)} \\ $$$$\:\:\:\:\:\:\:\:=\:{e}^{\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\:\left(−\mathrm{1}\:\right)^{\:{k}+\mathrm{1}} }{{k}}} \\ $$$$\:\:\:\:\:\:\:\:\:\therefore\:\:\mathrm{P}\:=\:{lim}_{\:{n}\rightarrow\infty} \left({e}^{\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\:{k}+\mathrm{1}} }{{k}}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{e}^{\:{lim}_{\:{n}\rightarrow\infty} \left(\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{e}^{\:{ln}\left(\mathrm{2}\right)} =\:\mathrm{2} \\ $$

Question Number 175766 Answers: 1 Comments: 1

Number of even composite factors of 2520?

$${Number}\:{of}\:\:{even}\:{composite}\:{factors}\:{of}\:\mathrm{2520}? \\ $$

Question Number 175765 Answers: 1 Comments: 0

lim_(x→∞) (1+(2/x))^(−x) =?

$${li}\underset{{x}\rightarrow\infty} {{m}}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}}\right)^{−{x}} =? \\ $$

Question Number 175762 Answers: 0 Comments: 0

lim_(x→0) ([((nsinx )/x)]+[((ntanx )/x)]) , where [:] denotes the greatest integer function and n∈I−{0}

Question Number 175758 Answers: 1 Comments: 0