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Question Number 213413    Answers: 2   Comments: 0

Find: A= [(√1)] + [(√2)] + [(√3) ]+...+ [(√(323))] = ?

$$\mathrm{Find}: \\ $$$$\mathrm{A}=\:\left[\sqrt{\mathrm{1}}\right]\:+\:\left[\sqrt{\mathrm{2}}\right]\:+\:\left[\sqrt{\mathrm{3}}\:\right]+...+\:\left[\sqrt{\mathrm{323}}\right]\:=\:? \\ $$

Question Number 213417    Answers: 2   Comments: 0

a , b , c , d ∈ N a + b + c + d = 63 Find: maksimum(ab + bc + cd) = ?

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:,\:\mathrm{d}\:\in\:\mathbb{N} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:+\:\mathrm{d}\:=\:\mathrm{63} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{maksimum}\left(\mathrm{ab}\:+\:\mathrm{bc}\:+\:\mathrm{cd}\right)\:=\:? \\ $$

Question Number 213404    Answers: 0   Comments: 3

pls teach me above question ↓↓ (prove real analysis pls) and sorry Mr gaster i cant believe you answer....

$$\mathrm{pls}\:\mathrm{teach}\:\mathrm{me}\:\mathrm{above}\:\mathrm{question} \\ $$$$\downarrow\downarrow\:\left(\mathrm{prove}\:\mathrm{real}\:\mathrm{analysis}\:\mathrm{pls}\right) \\ $$$$\mathrm{and}\:\mathrm{sorry}\:\mathrm{Mr}\:\mathrm{gaster} \\ $$$$\mathrm{i}\:\mathrm{cant}\:\mathrm{believe}\:\mathrm{you}\:\mathrm{answer}.... \\ $$

Question Number 213398    Answers: 2   Comments: 0

One simple Equation pls prove this property Σ_(j=1) ^N a_j ∙Σ_(k=1) ^M b_k =Σ_(j=1) ^N ∙Σ_(k=1) ^M a_j b_k .. and Σ_(j=0) ^N f(a+((b−a)/N)j)∙((b−a)/N)∙Σ_(k=0) ^M g(a+((b−a)/M)k)∙((b−a)/M) =Σ_(j=0) ^N Σ_(k=0) ^M f(a+((b−a)/N)j)g(a+((b−a)/M)k)(((b−a)^2 )/(MN)) But..... that Sum not euqal to ∫_a ^( b) f(z)g(z)dz... why integral form dosen′t work like Summation Σ_(j=1) ^N f(j) ∙Σ_(k=1) ^M g(k)=Σ_(j=1) ^N Σ_(k=1) ^M f(j)g(k) is True But..... ∫_a ^b f(u)du∙ ∫_a ^b g(v)dv isn′t equal to ∫_a ^b f(w)g(w)dw

$$\mathrm{One}\:\mathrm{simple}\:\mathrm{Equation} \\ $$$$\mathrm{pls}\:\mathrm{prove}\:\mathrm{this}\:\mathrm{property} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{a}_{{j}} \centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}{b}_{{k}} =\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:{a}_{{j}} {b}_{{k}} .. \\ $$$$\:\:\mathrm{and} \\ $$$$\underset{{j}=\mathrm{0}} {\overset{{N}} {\sum}}\:{f}\left({a}+\frac{{b}−{a}}{{N}}{j}\right)\centerdot\frac{{b}−{a}}{{N}}\centerdot\underset{{k}=\mathrm{0}} {\overset{{M}} {\sum}}\:\mathrm{g}\left({a}+\frac{{b}−{a}}{{M}}{k}\right)\centerdot\frac{{b}−{a}}{{M}} \\ $$$$=\underset{{j}=\mathrm{0}} {\overset{{N}} {\sum}}\:\underset{{k}=\mathrm{0}} {\overset{{M}} {\sum}}\:{f}\left({a}+\frac{{b}−{a}}{{N}}{j}\right)\mathrm{g}\left({a}+\frac{{b}−{a}}{{M}}{k}\right)\frac{\left({b}−{a}\right)^{\mathrm{2}} }{{MN}} \\ $$$$\mathrm{But}..... \\ $$$$\mathrm{that}\:\mathrm{Sum}\:\mathrm{not}\:\mathrm{euqal}\:\mathrm{to}\:\int_{{a}} ^{\:{b}} \:{f}\left({z}\right)\mathrm{g}\left({z}\right)\mathrm{d}{z}... \\ $$$$\mathrm{why}\:\mathrm{integral}\:\mathrm{form}\:\mathrm{dosen}'\mathrm{t}\:\mathrm{work} \\ $$$$\:\mathrm{like}\:\mathrm{Summation} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{f}\left({j}\right)\:\centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:\mathrm{g}\left({k}\right)=\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:{f}\left({j}\right)\mathrm{g}\left({k}\right) \\ $$$$\:\mathrm{is}\:\mathrm{True}\:\mathrm{But}..... \\ $$$$\int_{{a}} ^{{b}} \:{f}\left({u}\right)\mathrm{d}{u}\centerdot\:\int_{{a}} ^{{b}} \:\mathrm{g}\left({v}\right)\mathrm{d}{v}\:\:\mathrm{isn}'\mathrm{t}\:\mathrm{equal}\:\mathrm{to}\: \\ $$$$\int_{{a}} ^{{b}} \:{f}\left({w}\right)\mathrm{g}\left({w}\right)\mathrm{d}{w} \\ $$

Question Number 213397    Answers: 1   Comments: 0

Question Number 213391    Answers: 1   Comments: 0

Solve the system of equations where a,b,c≥0 a−2bc=b−2ac=c−2ab a+b+c=2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}\:\mathrm{where}\:{a},{b},{c}\geqslant\mathrm{0} \\ $$$${a}−\mathrm{2}{bc}={b}−\mathrm{2}{ac}={c}−\mathrm{2}{ab} \\ $$$${a}+{b}+{c}=\mathrm{2}\: \\ $$

Question Number 213330    Answers: 1   Comments: 1

Question Number 213323    Answers: 1   Comments: 1

Question Number 213322    Answers: 4   Comments: 1

Question Number 213298    Answers: 1   Comments: 0

Question 29. Theres 2 same-ratio sequences {a_n },{b_n } with a ratio of non-zero And if two sums of each sequences (Σ_(n=1) ^∞ a_n ,Σ_(n=1) ^∞ b_n ) are convergent, and two equations Σ_(n=1) ^∞ a_n b_n =(Σ_(n=1) ^∞ a_n )×(Σ_(n=1) ^∞ b_n ) and 3×Σ_(n=1) ^∞ ∣a_(2n) ∣=7×Σ_(n=1) ^∞ ∣a_(3n) ∣ are true. If Σ_(n=1) ^∞ ((b_(2n−1) +b_(3n+1) )/b_n )=S, then Find the value of 120S. (korea university exam question)

$$\boldsymbol{{Question}}\:\mathrm{29}.\:{Theres}\:\mathrm{2}\:{same}-{ratio}\:{sequences}\:\left\{{a}_{{n}} \right\},\left\{{b}_{{n}} \right\}\:{with}\:{a}\:{ratio}\:{of}\:{non}-{zero} \\ $$$${And}\:{if}\:{two}\:{sums}\:{of}\:{each}\:{sequences}\:\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} ,\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{b}_{{n}} \right)\:{are}\:{convergent},\:{and}\:{two}\:{equations} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} {b}_{{n}} =\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} \right)×\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{b}_{{n}} \right)\:{and}\:\mathrm{3}×\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mid{a}_{\mathrm{2}{n}} \mid=\mathrm{7}×\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mid{a}_{\mathrm{3}{n}} \mid\:{are}\:{true}. \\ $$$${If}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{b}_{\mathrm{2}{n}−\mathrm{1}} +{b}_{\mathrm{3}{n}+\mathrm{1}} }{{b}_{{n}} }={S},\:{then}\:\boldsymbol{{Find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\mathrm{120}\boldsymbol{{S}}. \\ $$$$\left({korea}\:{university}\:{exam}\:{question}\right) \\ $$

Question Number 213292    Answers: 0   Comments: 0

hey tinku tara I cant plot functions even i am logged in it says check if the variable name is “x” and you are logged in

$${hey}\:{tinku}\:{tara} \\ $$$${I}\:{cant}\:{plot}\:{functions} \\ $$$${even}\:{i}\:{am}\:{logged}\:{in} \\ $$$${it}\:{says}\:{check}\:{if}\:{the}\:{variable}\:{name}\:{is}\:``{x}''\:{and}\:{you}\:{are}\:{logged}\:{in} \\ $$

Question Number 213291    Answers: 1   Comments: 0

Find domain of y_(213291) : y_(213291) =((3+e^((x^2 −3x+2)/(x−6)) )/(log_(3/4) (√(x^2 −(1/4)))))

$${Find}\:{domain}\:{of}\:{y}_{\mathrm{213291}} : \\ $$$${y}_{\mathrm{213291}} =\frac{\mathrm{3}+{e}^{\frac{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}{{x}−\mathrm{6}}} }{\mathrm{log}_{\frac{\mathrm{3}}{\mathrm{4}}} \sqrt{{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}} \\ $$

Question Number 213290    Answers: 1   Comments: 0

for every real set R , f∈R and f is Smooth function. and f is f∈C^2 ∀_x f^((1)) (x)>0 , f^((2)) (x)<0 then prove ∣∫_0 ^( t) cos(f(x))dx∣≤(2/(f^((1)) (t))) t∈R

$$\mathrm{for}\:\mathrm{every}\:\mathrm{real}\:\mathrm{set}\:\mathbb{R}\:,\:{f}\in\mathbb{R} \\ $$$$\mathrm{and}\:{f}\:\mathrm{is}\:\mathrm{Smooth}\:\mathrm{function}.\:\mathrm{and}\:{f}\:\mathrm{is}\:{f}\in\mathcal{C}^{\mathrm{2}} \\ $$$$\forall_{{x}} \:{f}^{\left(\mathrm{1}\right)} \left({x}\right)>\mathrm{0}\:,\:{f}^{\left(\mathrm{2}\right)} \left({x}\right)<\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mid\int_{\mathrm{0}} ^{\:{t}} \:\mathrm{cos}\left({f}\left({x}\right)\right)\mathrm{d}{x}\mid\leq\frac{\mathrm{2}}{{f}^{\left(\mathrm{1}\right)} \left({t}\right)} \\ $$$${t}\in\mathbb{R} \\ $$

Question Number 213288    Answers: 0   Comments: 0

Question Number 213283    Answers: 1   Comments: 0

a_h is Cauchy Sequence. Sequence {a_h }_(h=1) ^n Satisfy Σ_(h=1) ^n a_h =0 , Σ_(h=1) ^n a_h ^2 =1 find minimum value of Summation a_1 a_n +Σ_(h=1) ^(n−1) a_h a_(h+1) (korea university math contest problem)

$${a}_{{h}} \:\mathrm{is}\:\mathrm{Cauchy}\:\mathrm{Sequence}. \\ $$$$\mathrm{Sequence}\:\left\{{a}_{{h}} \right\}_{{h}=\mathrm{1}} ^{{n}} \mathrm{Satisfy}\:\underset{{h}=\mathrm{1}} {\overset{{n}} {\sum}}\:{a}_{{h}} =\mathrm{0}\:,\:\underset{{h}=\mathrm{1}} {\overset{{n}} {\sum}}\:{a}_{{h}} ^{\mathrm{2}} =\mathrm{1} \\ $$$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{Summation} \\ $$$$\:{a}_{\mathrm{1}} {a}_{{n}} +\underset{{h}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\:{a}_{{h}} {a}_{{h}+\mathrm{1}} \\ $$$$\left(\mathrm{korea}\:\mathrm{university}\:\mathrm{math}\:\mathrm{contest}\:\mathrm{problem}\right) \\ $$

Question Number 213281    Answers: 2   Comments: 1

Question Number 213276    Answers: 1   Comments: 2

y^2 = − 4px At (−(1/3),1)→ 1= −4p (− (1/3) − h) At (−(5/3),2)→ 4 = −4p (− (5/3) − h) 4 = ((− (5/3) − h)/(− (1/3) − h)) (4/3) + 4h = (5/3) + h 3h = (1/3) h = (1/9)

$$\mathrm{y}^{\mathrm{2}} \:=\:−\:\mathrm{4px} \\ $$$$\:\mathrm{At}\:\left(−\frac{\mathrm{1}}{\mathrm{3}},\mathrm{1}\right)\rightarrow\:\mathrm{1}=\:−\mathrm{4p}\:\left(−\:\frac{\mathrm{1}}{\mathrm{3}}\:−\:\mathrm{h}\right) \\ $$$$\:\mathrm{At}\:\left(−\frac{\mathrm{5}}{\mathrm{3}},\mathrm{2}\right)\rightarrow\:\mathrm{4}\:=\:−\mathrm{4p}\:\left(−\:\frac{\mathrm{5}}{\mathrm{3}}\:−\:\mathrm{h}\right) \\ $$$$\:\:\:\:\:\mathrm{4}\:=\:\frac{−\:\frac{\mathrm{5}}{\mathrm{3}}\:−\:\mathrm{h}}{−\:\frac{\mathrm{1}}{\mathrm{3}}\:−\:\mathrm{h}} \\ $$$$\:\:\:\:\frac{\mathrm{4}}{\mathrm{3}}\:+\:\mathrm{4h}\:=\:\frac{\mathrm{5}}{\mathrm{3}}\:+\:\mathrm{h}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3h}\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{h}\:=\:\frac{\mathrm{1}}{\mathrm{9}} \\ $$

Question Number 213267    Answers: 1   Comments: 0

∫_0 ^(+∞) ((x−x^2 +x^3 −…−x^(2018) )/((1+x)^(2021) ))dx

$$ \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{+\infty} \frac{{x}−{x}^{\mathrm{2}} +{x}^{\mathrm{3}} −\ldots−{x}^{\mathrm{2018}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{2021}} }{dx} \\ $$

Question Number 213264    Answers: 0   Comments: 0

(√2) x^3 + x + 1 = 0 x_1 , x_2 □

$$\:\:\:\:\:\:\sqrt{\mathrm{2}}\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{x}\:+\:\mathrm{1}\:=\:\mathrm{0}\: \\ $$$$\:\:\:\:\:\:\mathrm{x}_{\mathrm{1}} \:,\:\mathrm{x}_{\mathrm{2}} \:\:\underline{\underbrace{\square}} \\ $$

Question Number 213261    Answers: 3   Comments: 0

if between 10^4 and 10^n there are 9999000 coprime numbers with 20. find n. please. thanks

$${if}\:{between}\:\:\mathrm{10}^{\mathrm{4}} \:{and}\:\mathrm{10}^{{n}} \:{there}\:{are}\:\mathrm{9999000}\:{coprime}\:{numbers}\:{with}\:\mathrm{20}.\:{find}\:{n}.\:\:{please}.\:{thanks} \\ $$

Question Number 213250    Answers: 3   Comments: 0

Question Number 213241    Answers: 3   Comments: 0

lim_(x→0) ((sin x−tan x)/x^3 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{x}−\mathrm{tan}\:{x}}{{x}^{\mathrm{3}} } \\ $$

Question Number 213234    Answers: 2   Comments: 2

Question Number 213232    Answers: 0   Comments: 4

Just a warning: the solutions of these two here are very often wrong: MrGaster lepuissantcedricjunior They also do not answer (my) comments regarding their errors. If you need the answers to these questions for an exam or other important reasons you might face serious problems.

$$\mathrm{Just}\:\mathrm{a}\:\mathrm{warning}:\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{these}\:\mathrm{two} \\ $$$$\mathrm{here}\:\mathrm{are}\:\mathrm{very}\:\mathrm{often}\:\mathrm{wrong}: \\ $$$$ \\ $$$$\mathrm{MrGaster} \\ $$$$\mathrm{lepuissantcedricjunior} \\ $$$$ \\ $$$$\mathrm{They}\:\mathrm{also}\:\mathrm{do}\:\mathrm{not}\:\mathrm{answer}\:\left(\mathrm{my}\right)\:\mathrm{comments} \\ $$$$\mathrm{regarding}\:\mathrm{their}\:\mathrm{errors}. \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{need}\:\mathrm{the}\:\mathrm{answers}\:\mathrm{to}\:\mathrm{these}\:\mathrm{questions} \\ $$$$\mathrm{for}\:\mathrm{an}\:\mathrm{exam}\:\mathrm{or}\:\mathrm{other}\:\mathrm{important}\:\mathrm{reasons} \\ $$$$\mathrm{you}\:\mathrm{might}\:\mathrm{face}\:\mathrm{serious}\:\mathrm{problems}. \\ $$

Question Number 213221    Answers: 1   Comments: 3

Question Number 213217    Answers: 1   Comments: 0

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