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AlgebraQuestion and Answers: Page 1

Question Number 209633    Answers: 2   Comments: 0

Question Number 209580    Answers: 2   Comments: 0

If a_n >0 and lim_(n→∞) a_n = 0 Find: lim_(n→∞) (1/n) Σ_(k=1) ^n ln ((k/n) + a_n ) = ?

$$\mathrm{If}\:\:\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} >\mathrm{0}\:\:\:\mathrm{and}\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:=\:\mathrm{0} \\ $$$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{ln}\:\left(\frac{\mathrm{k}}{\mathrm{n}}\:+\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \right)\:=\:? \\ $$

Question Number 209542    Answers: 2   Comments: 0

∫(dx/(x^(15) −x^(11) ))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{{x}^{\mathrm{15}} −{x}^{\mathrm{11}} } \\ $$$$ \\ $$$$ \\ $$

Question Number 209539    Answers: 0   Comments: 0

Question Number 209531    Answers: 0   Comments: 1

(y^(′′) /y) = 4x^2 + 2

$$\frac{\mathrm{y}^{''} }{\mathrm{y}}\:\:=\:\:\mathrm{4x}^{\mathrm{2}} \:\:+\:\:\mathrm{2} \\ $$

Question Number 209514    Answers: 1   Comments: 0

If Σ a_n is absolutely convergent, prove that Σ (a_n /n) is also absolutely convergent.

$$\mathrm{If}\:\Sigma\:{a}_{{n}} \:\mathrm{is}\:\mathrm{absolutely}\:\mathrm{convergent},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\Sigma\:\frac{{a}_{{n}} }{{n}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{absolutely}\:\mathrm{convergent}. \\ $$

Question Number 209456    Answers: 2   Comments: 0

Question Number 209455    Answers: 1   Comments: 0

Question Number 209436    Answers: 1   Comments: 2

{ ((x + y + z = 1)),((42x + 44y + 30z = 42)) :} (x,y,z)=(1,0,0) yes, but solution...

$$\begin{cases}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{1}}\\{\mathrm{42x}\:+\:\mathrm{44y}\:+\:\mathrm{30z}\:=\:\mathrm{42}}\end{cases} \\ $$$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\mathrm{1},\mathrm{0},\mathrm{0}\right)\:\mathrm{yes},\:\mathrm{but}\:\mathrm{solution}... \\ $$

Question Number 209415    Answers: 1   Comments: 1

Question Number 209404    Answers: 2   Comments: 0

Question Number 209385    Answers: 1   Comments: 0

Question Number 209357    Answers: 3   Comments: 0

Evaluate : B_n = Π_(k=3) ^n (( k^( 2) −1)/(k^2 + k −6))= ?

$$ \\ $$$$\:\:\:\:\:\:{Evaluate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{B}_{{n}} =\:\underset{{k}=\mathrm{3}} {\overset{{n}} {\prod}}\:\frac{\:{k}^{\:\mathrm{2}} −\mathrm{1}}{{k}^{\mathrm{2}} \:+\:{k}\:−\mathrm{6}}=\:? \\ $$

Question Number 209318    Answers: 2   Comments: 0

Question Number 209309    Answers: 0   Comments: 0

m , n ∈ N m ≥ 2 and n ≥ 2 p > 0 and q > 0 p + q = 1 Prove that: (1−q^n )^m + (1−p^m )^n ≥ 1

$$\mathrm{m}\:,\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{m}\:\geqslant\:\mathrm{2}\:\:\:\mathrm{and}\:\:\:\mathrm{n}\:\geqslant\:\mathrm{2} \\ $$$$\mathrm{p}\:>\:\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{q}\:>\:\mathrm{0} \\ $$$$\mathrm{p}\:+\:\mathrm{q}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\left(\mathrm{1}−\mathrm{q}^{\boldsymbol{\mathrm{n}}} \right)^{\boldsymbol{\mathrm{m}}} \:+\:\left(\mathrm{1}−\mathrm{p}^{\boldsymbol{\mathrm{m}}} \right)^{\boldsymbol{\mathrm{n}}} \:\geqslant\:\mathrm{1} \\ $$

Question Number 209290    Answers: 0   Comments: 1

a^2 −a−^(1000) (√((1+8000a)))=1000 find a

$$\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}−^{\mathrm{1000}} \sqrt{\left(\mathrm{1}+\mathrm{8000}\boldsymbol{\mathrm{a}}\right)}=\mathrm{1000} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{a}} \\ $$

Question Number 209263    Answers: 0   Comments: 5

6 different letters were written to 6 different people and 6 different envelopes were prepared with the addresses of these people written on them. In how many different ways can you put a letter in each envelope without putting a letter written to this person in the envelope with the name of any person?

$$ \\ $$6 different letters were written to 6 different people and 6 different envelopes were prepared with the addresses of these people written on them. In how many different ways can you put a letter in each envelope without putting a letter written to this person in the envelope with the name of any person?

Question Number 209240    Answers: 1   Comments: 0

If x + ((49)/(x + 48)) = − 34 find (2x + 83)^3 + (1/((2x + 83)^3 ))

$${If}\:\:{x}\:\:+\:\:\frac{\mathrm{49}}{{x}\:+\:\mathrm{48}}\:\:=\:\:−\:\mathrm{34} \\ $$$${find}\:\:\left(\mathrm{2}{x}\:+\:\mathrm{83}\right)^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\left(\mathrm{2}{x}\:+\:\mathrm{83}\right)^{\mathrm{3}} } \\ $$

Question Number 209234    Answers: 2   Comments: 0

Arrange in descending order: (√5) − (√2), (√7) − (√5) , (√(13)) − (√(11)) , (√(19)) − (√(17))

$$\mathrm{Arrange}\:\mathrm{in}\:\mathrm{descending}\:\mathrm{order}: \\ $$$$\:\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\sqrt{\mathrm{2}},\:\:\:\:\:\sqrt{\mathrm{7}}\:\:−\:\:\sqrt{\mathrm{5}}\:,\:\:\:\sqrt{\mathrm{13}}\:\:−\:\:\sqrt{\mathrm{11}}\:,\:\:\:\sqrt{\mathrm{19}}\:\:−\:\:\sqrt{\mathrm{17}} \\ $$

Question Number 209223    Answers: 2   Comments: 0

Question Number 209187    Answers: 3   Comments: 0

:: α , β and γ are roots of the following equation . Find the value of ” F ” : Equation : x^( 3) −2x −1=0 F := α^( 5) + β^( 5) + γ^( 5)

$$ \\ $$$$\:\:\:::\:\:\:\alpha\:,\:\beta\:\:{and}\:\:\gamma\:\:{are}\:{roots}\:{of}\:{the} \\ $$$$\:\:\:\:\:{following}\:\:{equation}\:.\:{Find}\:{the} \\ $$$$\:\:\:\:\:{value}\:\:{of}\:\:\:''\:\:\mathrm{F}\:\:''\::\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{E}{quation}\::\:\:\:\:\:\:{x}^{\:\mathrm{3}} \:−\mathrm{2}{x}\:\:−\mathrm{1}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{F}\::=\:\alpha^{\:\mathrm{5}} \:+\:\beta^{\:\mathrm{5}} \:+\:\gamma^{\:\mathrm{5}} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 209131    Answers: 1   Comments: 0

prove : curve { ((x(t)=((a+r.cos(t))/(a^2 +r^2 +2ar.cos(t))))),((y(t)=((r.sin(t))/(a^2 +r^2 +2ar.cos(t))))) :} 0≤t≤2π is circle , find center & radius

$${prove}\:: \\ $$$${curve}\:\begin{cases}{{x}\left({t}\right)=\frac{{a}+{r}.{cos}\left({t}\right)}{{a}^{\mathrm{2}} +{r}^{\mathrm{2}} +\mathrm{2}{ar}.{cos}\left({t}\right)}}\\{{y}\left({t}\right)=\frac{{r}.{sin}\left({t}\right)}{{a}^{\mathrm{2}} +{r}^{\mathrm{2}} +\mathrm{2}{ar}.{cos}\left({t}\right)}}\end{cases}\:\:\:\:\mathrm{0}\leqslant{t}\leqslant\mathrm{2}\pi \\ $$$${is}\:{circle}\:,\:{find}\:{center}\:\&\:{radius} \\ $$

Question Number 209072    Answers: 0   Comments: 8

Question Number 209065    Answers: 1   Comments: 0

The cost of maintaining a school is partly constant and partly varies as the number of students. With 50 students the cost is $15705 and with 40 students the cost is $13305. If the fee per student is $360.00, what is the least number of students for which the school can be run without loss?

$${The}\:{cost}\:{of}\:{maintaining}\:{a}\:{school}\:{is} \\ $$$${partly}\:{constant}\:{and}\:{partly}\:{varies}\:{as} \\ $$$${the}\:{number}\:{of}\:{students}.\:{With}\:\mathrm{50}\:{students} \\ $$$${the}\:{cost}\:{is}\:\$\mathrm{15705}\:{and}\:{with}\:\mathrm{40}\:{students} \\ $$$${the}\:{cost}\:{is}\:\$\mathrm{13305}.\:{If}\:{the}\:{fee}\:{per}\:{student} \\ $$$${is}\:\$\mathrm{360}.\mathrm{00},\:{what}\:{is}\:{the}\:{least}\:{number}\:{of} \\ $$$${students}\:{for}\:{which}\:{the}\:{school}\:{can}\:{be} \\ $$$${run}\:{without}\:{loss}? \\ $$

Question Number 209062    Answers: 1   Comments: 0

Question Number 209059    Answers: 0   Comments: 3

Compare: 8! and 8!!

$$\mathrm{Compare}: \\ $$$$\mathrm{8}!\:\:\:\mathrm{and}\:\:\:\mathrm{8}!! \\ $$

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