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Question Number 192450 Answers: 1 Comments: 0

The first term of an arithmetic series is 7 and the last is 70. and the sum is 385. find the number of terms in the series and its common difference.

$$\mathrm{The}\:\mathrm{first}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{arithmetic}\:\mathrm{series} \\ $$$$\mathrm{is}\:\mathrm{7}\:\mathrm{and}\:\mathrm{the}\:\mathrm{last}\:\mathrm{is}\:\mathrm{70}.\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{is} \\ $$$$\mathrm{385}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{series}\:\mathrm{and}\:\mathrm{its}\:\mathrm{common}\:\mathrm{difference}. \\ $$

Question Number 192444 Answers: 1 Comments: 4

The sum of three numbers in arith- metic progression is 18 and sum of square is 206. find the numbers

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{three}\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{arith}- \\ $$$$\mathrm{metic}\:\mathrm{progression}\:\mathrm{is}\:\mathrm{18}\:\mathrm{and}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{square}\:\mathrm{is}\:\mathrm{206}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{numbers} \\ $$

Question Number 192443 Answers: 1 Comments: 0

Find the sum of the first 16th term of the series 3(1/2) + 4(3/4) + 6 + 7(1/4) ...

$$\mathrm{Find}\:\mathrm{the}\:\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{16th}\:\mathrm{term} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{series}\:\mathrm{3}\frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{4}\frac{\mathrm{3}}{\mathrm{4}}\:+\:\mathrm{6}\:+\:\mathrm{7}\frac{\mathrm{1}}{\mathrm{4}}\:... \\ $$

Question Number 192440 Answers: 1 Comments: 0

Given { ((A=(((p^2 +q^2 +r^2 )^2 )/((pq)^2 +(pr)^2 +(qr)^2 )))),((B=((q^2 −pr)/(p^2 +q^2 +r^2 )) )) :} If p+q+r=0 then A^2 −4B=?

$$\:\:\:\:\mathrm{Given}\:\begin{cases}{\mathrm{A}=\frac{\left(\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left(\mathrm{pq}\right)^{\mathrm{2}} +\left(\mathrm{pr}\right)^{\mathrm{2}} +\left(\mathrm{qr}\right)^{\mathrm{2}} }}\\{\mathrm{B}=\frac{\mathrm{q}^{\mathrm{2}} −\mathrm{pr}}{\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} }\:}\end{cases}\:\:\:\:\:\: \\ $$$$\:\mathrm{If}\:\mathrm{p}+\mathrm{q}+\mathrm{r}=\mathrm{0}\:\mathrm{then}\:\mathrm{A}^{\mathrm{2}} −\mathrm{4B}=? \\ $$$$ \\ $$

Question Number 192437 Answers: 1 Comments: 0

(1/a) + (1/b) + (1/c) = (1/(a + b + c)) . Prove that (1/a^5 ) + (1/b^5 ) + (1/c^5 ) = (1/(a^5 + b^5 + c^5 )) = (1/((a + b + c)^5 ))

$$\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{{a}\:+\:{b}\:+\:{c}}\:.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{5}} }\:+\:\frac{\mathrm{1}}{{b}^{\mathrm{5}} }\:+\:\frac{\mathrm{1}}{{c}^{\mathrm{5}} }\:=\:\frac{\mathrm{1}}{{a}^{\mathrm{5}} \:+\:{b}^{\mathrm{5}} \:+\:{c}^{\mathrm{5}} }\:=\:\frac{\mathrm{1}}{\left({a}\:+\:{b}\:+\:{c}\right)^{\mathrm{5}} } \\ $$

Question Number 192433 Answers: 1 Comments: 0

lim_(x→0) ((1−cos (ln (3x+1)))/(3x^6 −tan (3x^2 )))=?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{ln}\:\left(\mathrm{3x}+\mathrm{1}\right)\right)}{\mathrm{3x}^{\mathrm{6}} −\mathrm{tan}\:\left(\mathrm{3x}^{\mathrm{2}} \right)}=? \\ $$

Question Number 192429 Answers: 0 Comments: 0

Question Number 192428 Answers: 0 Comments: 0

Question Number 192426 Answers: 1 Comments: 0

log(−10)(−10)=?

$${log}\left(−\mathrm{10}\right)\left(−\mathrm{10}\right)=? \\ $$

Question Number 192425 Answers: 1 Comments: 0

Question if “k” is odd & A=1^k +2^k +...+n^(k ) & B=1+2+...+n prove that : B ∣ A

$${Question} \\ $$$${if}\:\:``{k}''\:{is}\:{odd}\:\:\&\:{A}=\mathrm{1}^{{k}} +\mathrm{2}^{{k}} +...+{n}^{{k}\:\:} \:\&\:\:{B}=\mathrm{1}+\mathrm{2}+...+{n} \\ $$$${prove}\:{that}\:\::\:\:{B}\:\mid\:{A}\: \\ $$

Question Number 192420 Answers: 0 Comments: 1

Question Number 192409 Answers: 1 Comments: 0

Question Number 192407 Answers: 1 Comments: 0

find the value of the following integral χ = ∫_0 ^( ∞) (( ln^( 2) (x ))/(1+ x^( 2) )) dx = ? −−−−−−−−−

$$ \\ $$$$\:\:\:\:\:{find}\:\:{the}\:{value}\:{of}\:{the}\:{following}\:{integral} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\chi\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{ln}^{\:\mathrm{2}} \left({x}\:\right)}{\mathrm{1}+\:{x}^{\:\mathrm{2}} }\:{dx}\:=\:?\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:−−−−−−−−− \\ $$

Question Number 192406 Answers: 1 Comments: 0

∫ ((√(x^2 −4))/x) dx???

$$\int\:\frac{\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}{{x}}\:{dx}??? \\ $$

Question Number 192399 Answers: 1 Comments: 1

Algebra (1 ) G, is a group and o(G ) = p^( 2) . prove that G is an abelian group. hint: ( p is prime number ) −−−−−−−−−−−−−

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Algebra}\:\left(\mathrm{1}\:\right) \\ $$$$\:\:{G},\:{is}\:{a}\:{group}\:\:{and}\:\:\:{o}\left({G}\:\right)\:=\:{p}^{\:\mathrm{2}} \:. \\ $$$$\:\:\:{prove}\:{that}\:{G}\:{is}\:{an}\:{abelian}\:{group}. \\ $$$$\:\:\:{hint}:\:\:\left(\:{p}\:{is}\:{prime}\:{number}\:\:\right) \\ $$$$\:\:\:\:\:−−−−−−−−−−−−− \\ $$

Question Number 192398 Answers: 1 Comments: 0

Let f:R^3 →R be define by f(x, y, z) = 2x^2 −y+6xy−z^3 +3z. calculate the directional deriva− tive of the vector u=(2, 1, −3) help!

$$\mathrm{Let}\:\mathrm{f}:\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{define}\:\mathrm{by}\: \\ $$$$\mathrm{f}\left(\mathrm{x},\:\mathrm{y},\:\mathrm{z}\right)\:=\:\mathrm{2x}^{\mathrm{2}} −\mathrm{y}+\mathrm{6xy}−\mathrm{z}^{\mathrm{3}} +\mathrm{3z}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{directional}\:\mathrm{deriva}− \\ $$$$\mathrm{tive}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vector}\:\mathrm{u}=\left(\mathrm{2},\:\mathrm{1},\:−\mathrm{3}\right) \\ $$$$ \\ $$$$\mathrm{help}! \\ $$

Question Number 192397 Answers: 1 Comments: 0

Let f:D(f)⊆R^n →R^m let ′a′ be an interior point of Dom(f) and let ′u′ be any vector in R^n , when is a vector v∈R^m called the directional derivative of f at ′a′ along the line determine by u ? help!

$$\mathrm{Let}\:\mathrm{f}:\mathrm{D}\left(\mathrm{f}\right)\subseteq\mathbb{R}^{\mathrm{n}} \rightarrow\mathbb{R}^{\mathrm{m}} \\ $$$$\mathrm{let}\:'\mathrm{a}'\:\mathrm{be}\:\mathrm{an}\:\mathrm{interior}\:\mathrm{point}\:\mathrm{of}\:\mathrm{Dom}\left(\mathrm{f}\right) \\ $$$$\mathrm{and}\:\mathrm{let}\:'\mathrm{u}'\:\mathrm{be}\:\mathrm{any}\:\mathrm{vector}\:\mathrm{in}\:\mathbb{R}^{\mathrm{n}} ,\:\mathrm{when} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{v}\in\mathbb{R}^{\mathrm{m}} \:\mathrm{called}\:\mathrm{the}\:\mathrm{directional} \\ $$$$\mathrm{derivative}\:\mathrm{of}\:\mathrm{f}\:\mathrm{at}\:'\mathrm{a}'\:\mathrm{along}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{determine}\:\mathrm{by}\:\mathrm{u}\:? \\ $$$$ \\ $$$$\mathrm{help}! \\ $$

Question Number 192396 Answers: 1 Comments: 0

Question Number 192390 Answers: 1 Comments: 0

Give the function: x^2 −7x−8=0 have two roots x_(1 ) and x_2 No solving the function Find: x_1 ^3 +x_2 +2023

$${Give}\:{the}\:{function}: \\ $$$${x}^{\mathrm{2}} −\mathrm{7}{x}−\mathrm{8}=\mathrm{0} \\ $$$${have}\:{two}\:{roots}\:{x}_{\mathrm{1}\:} {and}\:{x}_{\mathrm{2}} \\ $$$${No}\:{solving}\:{the}\:{function}\: \\ $$$${Find}:\:{x}_{\mathrm{1}} ^{\mathrm{3}} +{x}_{\mathrm{2}} +\mathrm{2023} \\ $$

Question Number 192388 Answers: 1 Comments: 0

Question Number 192387 Answers: 1 Comments: 0

Simplify (√(2(1+(√(4+(((2017^4 −1)/(2017^2 )))^2 ))))) is ....

$$\:\mathrm{Simplify}\: \\ $$$$\:\sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{4}+\left(\frac{\mathrm{2017}^{\mathrm{4}} −\mathrm{1}}{\mathrm{2017}^{\mathrm{2}} }\right)^{\mathrm{2}} }\right)}\: \\ $$$$\:\mathrm{is}\:....\: \\ $$

Question Number 192377 Answers: 1 Comments: 0

The probability density function f(x) of a variable x is given by f(x)= { ((kxsin πx 0≤x≤1)),((0 for all value of x)) :} Show that k=π and deduce that mean and the variance of the distribution are (1−(4/π^2 )) and (2/π^2 )(1−(8/π^2 ))

$${The}\:{probability}\:{density}\:{function}\:{f}\left({x}\right) \\ $$$$\:{of}\:{a}\:{variable}\:{x}\:{is}\:{given}\:{by}\: \\ $$$${f}\left({x}\right)=\begin{cases}{{kx}\mathrm{sin}\:\pi{x}\:\:\:\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}}\\{\mathrm{0}\:{for}\:{all}\:{value}\:{of}\:{x}}\end{cases} \\ $$$${Show}\:{that}\:{k}=\pi\:\:{and}\:{deduce}\:{that}\:{mean}\: \\ $$$${and}\:{the}\:{variance}\:{of}\:{the}\:{distribution}\:{are} \\ $$$$\left(\mathrm{1}−\frac{\mathrm{4}}{\pi^{\mathrm{2}} }\right)\:{and}\:\frac{\mathrm{2}}{\pi^{\mathrm{2}} }\left(\mathrm{1}−\frac{\mathrm{8}}{\pi^{\mathrm{2}} }\right) \\ $$

Question Number 192376 Answers: 0 Comments: 0

Find the first four moment of the binomial distribution

$${Find}\:{the}\:{first}\:{four}\:{moment}\:{of}\:{the} \\ $$$${binomial}\:{distribution} \\ $$$$ \\ $$

Question Number 192375 Answers: 1 Comments: 0

Show that E(Z)=0 and Var(Z)=1 where Z is the standard normal variable

$${Show}\:{that}\:{E}\left({Z}\right)=\mathrm{0}\:\:\:{and}\:{Var}\left({Z}\right)=\mathrm{1}\:{where} \\ $$$${Z}\:{is}\:{the}\:{standard}\:{normal}\:{variable} \\ $$

Question Number 192374 Answers: 3 Comments: 1

why “ 200!<100^(200) ” ?

$${why}\:\:\:``\:\mathrm{200}!<\mathrm{100}^{\mathrm{200}} \:''\:? \\ $$

Question Number 192370 Answers: 2 Comments: 0

Solve the equation x^4 − 2x^3 + 4x^2 + 6x − 21 = 0, Given that the sum of two of its roots is zero

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}\:\:\:\mathrm{x}^{\mathrm{4}} \:\:−\:\:\mathrm{2x}^{\mathrm{3}} \:\:+\:\:\mathrm{4x}^{\mathrm{2}} \:\:+\:\:\mathrm{6x}\:\:\:−\:\:\mathrm{21}\:\:\:=\:\:\:\mathrm{0},\:\: \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{of}\:\mathrm{its}\:\mathrm{roots}\:\mathrm{is}\:\mathrm{zero} \\ $$

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