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Question Number 157967 Answers: 0 Comments: 0

Prove that 7^(n+1) divide 3^7^n + 5^7^n - 1 for any positive integers n

$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{7}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:\:\mathrm{divide}\:\:\mathrm{3}^{\mathrm{7}^{\boldsymbol{\mathrm{n}}} } \:+\:\mathrm{5}^{\mathrm{7}^{\boldsymbol{\mathrm{n}}} } \:-\:\mathrm{1} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{integers}\:\:\boldsymbol{\mathrm{n}} \\ $$

Question Number 157965 Answers: 1 Comments: 1

Question Number 157964 Answers: 0 Comments: 1

Solve for real numbers: 4tan(x) + ((sin(5x))/(cos^5 (x))) = 0

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{4tan}\left(\mathrm{x}\right)\:+\:\frac{\mathrm{sin}\left(\mathrm{5x}\right)}{\mathrm{cos}^{\mathrm{5}} \left(\mathrm{x}\right)}\:=\:\mathrm{0} \\ $$$$ \\ $$

Question Number 157961 Answers: 2 Comments: 0

prove that: I=∫_0 ^( ∞) x^( 2) tanh(x).e^( −x) dx=(π^( 3) /8) −2

$$ \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}: \\ $$$$ \\ $$$$\mathrm{I}=\int_{\mathrm{0}} ^{\:\infty} {x}^{\:\mathrm{2}} {tanh}\left({x}\right).{e}^{\:−{x}} {dx}=\frac{\pi^{\:\mathrm{3}} }{\mathrm{8}}\:−\mathrm{2}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 157957 Answers: 1 Comments: 3

Question Number 157958 Answers: 0 Comments: 0

Find: Ω_n =Σ_(n=1) ^∞ (1/n)(Σ_(k=1) ^n ((k^3 + k^2 - 3k - 2)/((k + 2)!)))

$$\mathrm{Find}: \\ $$$$\Omega_{\boldsymbol{\mathrm{n}}} =\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n}}\left(\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\frac{\mathrm{k}^{\mathrm{3}} \:+\:\mathrm{k}^{\mathrm{2}} \:-\:\mathrm{3k}\:-\:\mathrm{2}}{\left(\mathrm{k}\:+\:\mathrm{2}\right)!}\right) \\ $$$$ \\ $$

Question Number 157952 Answers: 1 Comments: 0

f(x)=x^(2014) +2x^(2013) +3x^(2012) +4x^(2011) +...+2014x+2015 min f(x)=?

$$\:{f}\left({x}\right)={x}^{\mathrm{2014}} +\mathrm{2}{x}^{\mathrm{2013}} +\mathrm{3}{x}^{\mathrm{2012}} +\mathrm{4}{x}^{\mathrm{2011}} +...+\mathrm{2014}{x}+\mathrm{2015} \\ $$$$\:{min}\:{f}\left({x}\right)=? \\ $$

Question Number 157940 Answers: 0 Comments: 3

Question Number 158039 Answers: 2 Comments: 0

f (((x + 1)/2)) = x + 2 ⇒ f(x) = ?

$$\mathrm{f}\:\left(\frac{\mathrm{x}\:+\:\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{x}\:+\:\mathrm{2}\:\:\Rightarrow\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$

Question Number 157938 Answers: 0 Comments: 0

Question Number 157937 Answers: 0 Comments: 0

Question Number 157935 Answers: 0 Comments: 0

Evaluate A=Σ_(k=1) ^n (tan((kπ)/(2n)))

$$\:\mathrm{Evaluate}\:\boldsymbol{\mathrm{A}}=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\mathrm{tan}\frac{\mathrm{k}\pi}{\mathrm{2n}}\right) \\ $$

Question Number 157932 Answers: 0 Comments: 4

prove that tan^(−1) (((xy)/(rz)))+tan^(−1) (((xz)/(ry)))+tan^(−1) (((yz)/(rx)))=(π/2)

$${prove}\:{that}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{{xy}}{{rz}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{xz}}{{ry}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{yz}}{{rx}}\right)=\frac{\pi}{\mathrm{2}} \\ $$

Question Number 157929 Answers: 0 Comments: 1

∫(dx/(sin x+ sec x)) using wiestress substitution

$$\int\frac{\mathrm{dx}}{\mathrm{sin}\:\mathrm{x}+\:\mathrm{sec}\:\mathrm{x}} \\ $$$$\mathrm{using}\:\mathrm{wiestress}\:\mathrm{substitution} \\ $$

Question Number 157942 Answers: 0 Comments: 1

Question Number 157913 Answers: 1 Comments: 0

5^2 ∙ 5^4 ∙ 5^6 ∙ ... ∙ 5^(2x) = 0,04^(-28) find x=?

$$\mathrm{5}^{\mathrm{2}} \:\centerdot\:\mathrm{5}^{\mathrm{4}} \:\centerdot\:\mathrm{5}^{\mathrm{6}} \:\centerdot\:...\:\centerdot\:\mathrm{5}^{\mathrm{2}\boldsymbol{\mathrm{x}}} \:=\:\mathrm{0},\mathrm{04}^{-\mathrm{28}} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 157908 Answers: 1 Comments: 0

(a_n ) in numerical series 7+9+11+13+...+(2n+1)=an^2 +bn+c find a+b+c=?

$$\left(\mathrm{a}_{\boldsymbol{\mathrm{n}}} \right)\:\mathrm{in}\:\mathrm{numerical}\:\mathrm{series} \\ $$$$\mathrm{7}+\mathrm{9}+\mathrm{11}+\mathrm{13}+...+\left(\mathrm{2n}+\mathrm{1}\right)=\mathrm{an}^{\mathrm{2}} +\mathrm{bn}+\mathrm{c} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}=? \\ $$

Question Number 157906 Answers: 0 Comments: 4

f(x)=(px+1)(2x+q+4) function is a single function, find p+q+pq=?

$$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{px}+\mathrm{1}\right)\left(\mathrm{2x}+\mathrm{q}+\mathrm{4}\right)\:\mathrm{function}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{single}\:\mathrm{function},\:\mathrm{find}\:\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}}+\boldsymbol{\mathrm{pq}}=? \\ $$$$ \\ $$

Question Number 157926 Answers: 2 Comments: 0

if the line px+qy=r tangents the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1, then 1) prove a^2 p^2 +b^2 q^2 =r^2 2) find the coordinates of the touching point.

$${if}\:{the}\:{line}\:{px}+{qy}={r}\:{tangents}\:{the} \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1},\:{then}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:\boldsymbol{{a}}^{\mathrm{2}} \boldsymbol{{p}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} \boldsymbol{{q}}^{\mathrm{2}} =\boldsymbol{{r}}^{\mathrm{2}} \: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{coordinates}\:{of}\:{the}\: \\ $$$$\:\:\:\:\:{touching}\:{point}. \\ $$

Question Number 157925 Answers: 1 Comments: 0

Question Number 157947 Answers: 1 Comments: 0

What are the coordinates of the points on the curve x^2 −y^2 =16 which nearest to (0,6)?

$${What}\:{are}\:{the}\:{coordinates}\:{of}\:{the} \\ $$$${points}\:{on}\:{the}\:{curve}\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{16} \\ $$$${which}\:{nearest}\:{to}\:\left(\mathrm{0},\mathrm{6}\right)? \\ $$

Question Number 157891 Answers: 2 Comments: 0

Question Number 157887 Answers: 1 Comments: 2

Question Number 157884 Answers: 1 Comments: 0

(1/(sin(10°))) - 4 sin(70°) = ?

$$\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{10}°\right)}\:-\:\mathrm{4}\:\mathrm{sin}\left(\mathrm{70}°\right)\:=\:? \\ $$

Question Number 157883 Answers: 1 Comments: 0

lim_(n→∞) ((1/(n^2 +1)) + (2/(n^2 +1)) + ... + ((n-1)/(n^2 +1))) = ?

$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\:+\:\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\:+\:...\:+\:\frac{\mathrm{n}-\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\right)\:=\:? \\ $$

Question Number 157880 Answers: 0 Comments: 1

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