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Question Number 216995 Answers: 3 Comments: 0

Find all prime numbers p and q such that p^2 − q^2 = 2024

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{numbers}\:\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\: \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{p}^{\mathrm{2}} −\:\:\mathrm{q}^{\mathrm{2}} =\:\:\mathrm{2024} \\ $$

Question Number 217015 Answers: 2 Comments: 0

If f(2x + 1) = 3x + 5 Find f(x) = ?

$$\mathrm{If} \\ $$$$\mathrm{f}\left(\mathrm{2x}\:+\:\mathrm{1}\right)\:=\:\mathrm{3x}\:+\:\mathrm{5} \\ $$$$\mathrm{Find} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$

Question Number 216990 Answers: 2 Comments: 0

(1) ∫(sec^2 x∙(√(tan x)))dx=?

$$\left(\mathrm{1}\right)\:\int\left(\mathrm{sec}^{\mathrm{2}} {x}\centerdot\sqrt{\mathrm{tan}\:{x}}\right){dx}=? \\ $$

Question Number 216983 Answers: 1 Comments: 0

Question Number 217075 Answers: 0 Comments: 3

Help me please... $\vv{AB}$ and $\vv{CD}$ are two vectors, and $\vv{AB}$ is not the zero vector. Prove that if the vectors $\vv{AB}$ and $\vv{CD}$ are colinear, then there exists a real number $ k $ such that $ \vv{CD} = k \vv{AB} $. (don't use coordinates !)

$$ \\ $$Help me please... $\vv{AB}$ and $\vv{CD}$ are two vectors, and $\vv{AB}$ is not the zero vector. Prove that if the vectors $\vv{AB}$ and $\vv{CD}$ are colinear, then there exists a real number $ k $ such that $ \vv{CD} = k \vv{AB} $. (don't use coordinates !)

Question Number 216959 Answers: 0 Comments: 0

Question Number 216958 Answers: 3 Comments: 0

Question Number 216952 Answers: 0 Comments: 0

Question Number 216953 Answers: 0 Comments: 1

Question Number 216926 Answers: 1 Comments: 0

Evaluate 5^2 Σ_(n=1) ^∞ (1/2)(Σ_(m=2) ^∞ (2/(m^2 +2m)))^(n−1)

$${Evaluate}\:\mathrm{5}^{\mathrm{2}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}}\left(\underset{{m}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{2}}{{m}^{\mathrm{2}} +\mathrm{2}{m}}\right)^{{n}−\mathrm{1}} \\ $$

Question Number 216925 Answers: 1 Comments: 0

Question Number 216934 Answers: 0 Comments: 0

see q#216900

$${see}\:\underline{{q}#\mathrm{216900}} \\ $$

Question Number 216919 Answers: 2 Comments: 0

Question Number 216918 Answers: 1 Comments: 0

give a recurrence relation for I_n . I_n =∫_0 ^1 (x^n /(x+3))dx, ∀n ∈ N.

$${give}\:{a}\:{recurrence}\:{relation}\:{for}\:{I}_{{n}} . \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{{x}+\mathrm{3}}{dx},\:\forall{n}\:\in\:\mathbb{N}. \\ $$

Question Number 216917 Answers: 1 Comments: 0

lim_(x→+∞) ((√(x+(√(x+(√(x+(√x)))))))−(√x))

$${li}\underset{{x}\rightarrow+\infty} {{m}}\:\left(\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}}−\sqrt{{x}}\right) \\ $$

Question Number 216914 Answers: 2 Comments: 0

Prove that: (1/(1001)) + (1/(1002)) + ... + (1/(2000)) > (5/8)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}}{\mathrm{1001}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1002}}\:\:+\:\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{2000}}\:\:>\:\:\frac{\mathrm{5}}{\mathrm{8}} \\ $$

Question Number 216913 Answers: 1 Comments: 0

6(1/4)%

$$\mathrm{6}\frac{\mathrm{1}}{\mathrm{4}}\%\: \\ $$

Question Number 216912 Answers: 0 Comments: 0

Find all three-digit numbers n such that 1. n is divisible by the sum of its digits. 2. n is a perfect square.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}-\mathrm{digit}\:\mathrm{numbers}\:{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{1}.\:{n}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\:\mathrm{of}\:\:\mathrm{its}\:\:\mathrm{digits}. \\ $$$$\mathrm{2}.\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$

Question Number 216911 Answers: 1 Comments: 0

Find all positive integer x,y such that x^2 + y^2 + xy = 169

$${Find}\:{all}\:{positive}\:{integer}\:\mathrm{x},\mathrm{y}\:{such}\:{that} \\ $$$$\mathrm{x}^{\mathrm{2}} +\:\mathrm{y}^{\mathrm{2}} +\:\mathrm{xy}\:=\:\mathrm{169} \\ $$

Question Number 216910 Answers: 0 Comments: 0

Ed:.06 a function δ(x) is a composite function which is as follow [{(f ○ g)(x)} ○ {(g ○ f)(x)}] ○ [{(f ′ ○ g ′)(x)} ○ {(g ′ ○ f ′)(x)}] where f(x) = Π_(n = 1) ^∞ ((nx^3 − nx^2 − nx −n)/(n^3 x − n^2 x − nx −x)) g(x)= f ′′(x) ∫_( ψ) ^( δ) δ(x) dx ∈ R\Q ? true or false?

$${Ed}:.\mathrm{06} \\ $$$${a}\:{function}\:\delta\left({x}\right)\:{is}\:{a}\:{composite}\:{function} \\ $$$${which}\:{is}\:{as}\:{follow}\: \\ $$$$\left[\left\{\left({f}\:\circ\:{g}\right)\left({x}\right)\right\}\:\circ\:\left\{\left({g}\:\circ\:{f}\right)\left({x}\right)\right\}\right]\:\circ\:\left[\left\{\left({f}\:'\:\circ\:{g}\:'\right)\left({x}\right)\right\}\:\circ\:\left\{\left({g}\:'\:\circ\:{f}\:'\right)\left({x}\right)\right\}\right] \\ $$$${where} \\ $$$${f}\left({x}\right)\:=\:\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{{nx}^{\mathrm{3}} \:−\:{nx}^{\mathrm{2}} \:−\:{nx}\:\:−{n}}{{n}^{\mathrm{3}} {x}\:−\:{n}^{\mathrm{2}} {x}\:−\:{nx}\:−{x}} \\ $$$${g}\left({x}\right)=\:{f}\:''\left({x}\right) \\ $$$$\underset{\:\psi} {\int}\overset{\:\delta} {\:}\delta\left({x}\right)\:{dx}\:\in\:\:\mathbb{R}\backslash\mathbb{Q}\:? \\ $$$${true}\:{or}\:{false}? \\ $$

Question Number 216907 Answers: 1 Comments: 0

Prove:n!=1+Σ_(k=1) ^∞ (k^n /e^k )−Σ_(k=1) ^∞ ((B_k sin(πk)(n−k)!)/(πk))

$$\mathrm{Prove}:{n}!=\mathrm{1}+\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{k}^{{n}} }{{e}^{{k}} }−\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{B}_{{k}} \mathrm{sin}\left(\pi{k}\right)\left({n}−{k}\right)!}{\pi{k}} \\ $$

Question Number 216906 Answers: 0 Comments: 0

Prove:Γ(x)=(x^x /((2π)^(x−1) ))Π_(k=1) ^∞ (k^(2(x−1)) /(Π_(i=1) ^(x−1) [k^2 −((i/2))^2 ]))

$$\mathrm{Prove}:\Gamma\left({x}\right)=\frac{{x}^{{x}} }{\left(\mathrm{2}\pi\right)^{{x}−\mathrm{1}} }\underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\frac{{k}^{\mathrm{2}\left({x}−\mathrm{1}\right)} }{\underset{{i}=\mathrm{1}} {\overset{{x}−\mathrm{1}} {\prod}}\left[{k}^{\mathrm{2}} −\left(\frac{{i}}{\mathrm{2}}\right)^{\mathrm{2}} \right]} \\ $$

Question Number 216900 Answers: 0 Comments: 1

Question Number 216886 Answers: 0 Comments: 1

Evaluate ((Σ_(k=1) ^(10) (∫_0 ^k (4u+1)du))/(5^2 Σ_(n=1) ^∞ (1/2)(Σ_(n=2) ^∞ (2/(m^2 +2m)))^(n−1) ))∫_(sin^(−1) (((−(√2))/2))) ^((π/2)cos(π/2)) (((1−secθsinθ)/((tanθ+cotθ)/(ϱ^θ −ϱ^(πi) ))))dθ

$${Evaluate}\:\frac{\underset{{k}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left(\int_{\mathrm{0}} ^{{k}} \left(\mathrm{4}{u}+\mathrm{1}\right){du}\right)}{\mathrm{5}^{\mathrm{2}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}}\left(\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{2}}{{m}^{\mathrm{2}} +\mathrm{2}{m}}\right)^{{n}−\mathrm{1}} }\int_{{sin}^{−\mathrm{1}} \left(\frac{−\sqrt{\mathrm{2}}}{\mathrm{2}}\right)} ^{\frac{\pi}{\mathrm{2}}{cos}\frac{\pi}{\mathrm{2}}} \left(\frac{\mathrm{1}−{sec}\theta{sin}\theta}{\frac{{tan}\theta+{cot}\theta}{\varrho^{\theta} −\varrho^{\pi{i}} }}\right){d}\theta \\ $$

Question Number 216887 Answers: 0 Comments: 0

Π_(k=1) ^n cos((x/2^k ))=Pn(x) evaluate Pn(x) and P_n (x^2 +1)

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)={Pn}\left({x}\right) \\ $$$${evaluate}\:{Pn}\left({x}\right)\:\:{and}\:\:{P}_{{n}} \left({x}^{\mathrm{2}} +\mathrm{1}\right) \\ $$$$ \\ $$$$ \\ $$

Question Number 216875 Answers: 1 Comments: 0

Let p be a prime number greater than 3. Prove that p^2 − 1 is always divisible by 24.

$$\mathrm{Let}\:\:\mathrm{p}\:\:\mathrm{be}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{p}^{\mathrm{2}} −\:\mathrm{1}\:\: \\ $$$$\mathrm{is}\:\:\mathrm{always}\:\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{24}. \\ $$

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