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Question Number 111799 Answers: 3 Comments: 4

Question Number 111788 Answers: 1 Comments: 0

Question Number 111781 Answers: 0 Comments: 3

Question Number 111774 Answers: 1 Comments: 0

Question Number 111772 Answers: 1 Comments: 0

calculate lim_(n→+∞) Π_(k=1) ^n (1+((√(k(n−k)))/n^2 ))

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\left(\mathrm{1}+\frac{\sqrt{\mathrm{k}\left(\mathrm{n}−\mathrm{k}\right)}}{\mathrm{n}^{\mathrm{2}} }\right) \\ $$

Question Number 111771 Answers: 0 Comments: 0

calculate lim_(n→+∞) a_n ∫_0 ^1 x^(2n) sin(((πx)/2))dx with a_n =Σ_(k=1) ^n sin(((πk)/(2n)))

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{a}_{\mathrm{n}} \int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{2n}} \mathrm{sin}\left(\frac{\pi\mathrm{x}}{\mathrm{2}}\right)\mathrm{dx}\:\mathrm{with}\:\mathrm{a}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{sin}\left(\frac{\pi\mathrm{k}}{\mathrm{2n}}\right) \\ $$

Question Number 111770 Answers: 0 Comments: 0

f function continue on [0,1] find lim_(n→+∞) (1/n)Σ_(k=0) ^n (n−k)∫_(k/n) ^((k+1)/n) f(x)dx

$$\mathrm{f}\:\mathrm{function}\:\mathrm{continue}\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{\mathrm{n}}\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{n}−\mathrm{k}\right)\int_{\frac{\mathrm{k}}{\mathrm{n}}} ^{\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 111769 Answers: 0 Comments: 0

let u_n =Σ_(k=1) ^n (1/(√((k+n)(k+n+1)))) detremine lim_(n→+∞) u_n

$$\mathrm{let}\:\mathrm{u}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\sqrt{\left(\mathrm{k}+\mathrm{n}\right)\left(\mathrm{k}+\mathrm{n}+\mathrm{1}\right)}} \\ $$$$\mathrm{detremine}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{u}_{\mathrm{n}} \\ $$

Question Number 111768 Answers: 0 Comments: 0

explicite f(x) =∫_0 ^(2π) ln(x^2 −2xcosθ +1)dθ (x≠+^− 1)

$$\mathrm{explicite}\:\mathrm{f}\left(\mathrm{x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)\mathrm{d}\theta\:\:\:\:\:\:\:\:\left(\mathrm{x}\neq\overset{−} {+}\mathrm{1}\right) \\ $$

Question Number 111766 Answers: 2 Comments: 0

calculate lim_(n→+∞) Σ_(k=1) ^n ln((n/(n+k)))^(1/n)

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \mathrm{ln}\left(\frac{\mathrm{n}}{\mathrm{n}+\mathrm{k}}\right)^{\frac{\mathrm{1}}{\mathrm{n}}} \\ $$

Question Number 111762 Answers: 1 Comments: 0

caoculate ∫_0 ^(π/4) ln(1+2tanx)dx

$$\mathrm{caoculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{1}+\mathrm{2tanx}\right)\mathrm{dx} \\ $$

Question Number 111760 Answers: 0 Comments: 0

find lim_(x→1^+ ) ∫_x ^x^2 ((ln(t))/((t−1)^2 ))dx

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{1}^{+} } \:\:\:\int_{\mathrm{x}} ^{\mathrm{x}^{\mathrm{2}} } \:\:\frac{\mathrm{ln}\left(\mathrm{t}\right)}{\left(\mathrm{t}−\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 111756 Answers: 0 Comments: 0

find I_n =∫_0 ^(π/4) (du/(cos^n u))

$$\mathrm{find}\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{du}}{\mathrm{cos}^{\mathrm{n}} \mathrm{u}} \\ $$

Question Number 111755 Answers: 2 Comments: 0

calculate lim_(n→+∞) Σ_(k=1) ^n (√((n−k)/(n^3 +n^2 k)))

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\:\sqrt{\frac{\mathrm{n}−\mathrm{k}}{\mathrm{n}^{\mathrm{3}} \:+\mathrm{n}^{\mathrm{2}} \mathrm{k}}} \\ $$

Question Number 111754 Answers: 2 Comments: 0

calculate lim_(n→+∞) Σ_(k=1) ^(2n) (k/(k^2 +n^2 ))

$$ \\ $$$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{2n}} \:\:\:\:\frac{\mathrm{k}}{\mathrm{k}^{\mathrm{2}} \:+\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 111887 Answers: 0 Comments: 0

Question Number 111882 Answers: 1 Comments: 0

(√(bemath)) (1)lim_(x→∞) [(x^2 /(x+1)) − (x^2 /(x+3)) ] ? (2) prove that n^2 ≤ 2^n for ∀n∈N by mathematical induction

$$\:\:\sqrt{{bemath}} \\ $$$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left[\frac{{x}^{\mathrm{2}} }{{x}+\mathrm{1}}\:−\:\frac{{x}^{\mathrm{2}} }{{x}+\mathrm{3}}\:\right]\:? \\ $$$$\left(\mathrm{2}\right)\:{prove}\:{that}\:{n}^{\mathrm{2}} \:\leqslant\:\mathrm{2}^{{n}} \:{for}\:\forall{n}\in\mathbb{N} \\ $$$${by}\:{mathematical}\:{induction} \\ $$

Question Number 111745 Answers: 0 Comments: 6

Question Number 111826 Answers: 0 Comments: 2

There are two cards; one is yellow on both sides and the other one is yellow in one side and red on the other. The cards have the same probability ((1/2)) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is yellow, then the probability that the under side is also yellow is.

Question Number 111734 Answers: 1 Comments: 0

A chord which is a perpendicular bisector of radius of length 18cm in a circle, has length.

$$\mathrm{A}\:\mathrm{chord}\:\mathrm{which}\:\mathrm{is}\:\mathrm{a}\:\mathrm{perpendicular} \\ $$$$\mathrm{bisector}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{length}\:\mathrm{18cm}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{circle},\:\mathrm{has}\:\mathrm{length}. \\ $$$$ \\ $$

Question Number 111732 Answers: 1 Comments: 0

A blind man is to place 5 letters into 5 pigeon holes, how many ways can 4 of the letters be wrongly placed? (note that only one letter must be in a pigeon hole)

$$\mathrm{A}\:\mathrm{blind}\:\mathrm{man}\:\mathrm{is}\:\mathrm{to}\:\mathrm{place}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{into}\:\mathrm{5} \\ $$$$\mathrm{pigeon}\:\mathrm{holes},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{4}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{letters}\:\mathrm{be}\:\mathrm{wrongly}\:\mathrm{placed}? \\ $$$$\left(\mathrm{note}\:\mathrm{that}\:\mathrm{only}\:\mathrm{one}\:\mathrm{letter}\:\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{pigeon}\:\mathrm{hole}\right) \\ $$

Question Number 111730 Answers: 0 Comments: 9

How many triples of positive integers (x,y,z) satisfy 79x+80y+81z =2016

$$\mathrm{How}\:\mathrm{many}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\:\mathrm{satisfy}\: \\ $$$$\mathrm{79x}+\mathrm{80y}+\mathrm{81z}\:=\mathrm{2016} \\ $$

Question Number 112813 Answers: 0 Comments: 2

A binary operation has the property a∗(b∗c) = (a∗b)•c and that a∗a=1 for all non−zero real numbers a,b and c. (′•′ here represent multiplication). The solution of the equation 2016∗(6∗x)=100 can be written as (p/q) where p and q are relatively prime positive integers. What is q−p?

$$\mathrm{A}\:\mathrm{binary}\:\mathrm{operation}\:\mathrm{has}\:\mathrm{the}\:\mathrm{property} \\ $$$$\mathrm{a}\ast\left(\mathrm{b}\ast\mathrm{c}\right)\:=\:\left(\mathrm{a}\ast\mathrm{b}\right)\bullet\mathrm{c}\:\mathrm{and}\:\mathrm{that}\:\mathrm{a}\ast\mathrm{a}=\mathrm{1}\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{non}−\mathrm{zero}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}. \\ $$$$\left('\bullet'\:\mathrm{here}\:\mathrm{represent}\:\mathrm{multiplication}\right). \\ $$$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{2016}\ast\left(\mathrm{6}\ast\mathrm{x}\right)=\mathrm{100}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\frac{\mathrm{p}}{\mathrm{q}} \\ $$$$\mathrm{where}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{are}\:\mathrm{relatively}\:\mathrm{prime} \\ $$$$\mathrm{positive}\:\mathrm{integers}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{q}−\mathrm{p}? \\ $$

Question Number 112812 Answers: 0 Comments: 2

There are 2016 straight lines drawn on a board such that (1/2) of the lines are parallel to one another. (3/8) of them meet at a point and each of the remaining ones intersect with all other lines on the board. Determine the total number of intersections possible.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{2016}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{drawn}\:\mathrm{on} \\ $$$$\mathrm{a}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lines}\:\mathrm{are} \\ $$$$\mathrm{parallel}\:\mathrm{to}\:\mathrm{one}\:\mathrm{another}.\:\frac{\mathrm{3}}{\mathrm{8}}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{meet}\:\mathrm{at}\:\mathrm{a}\:\mathrm{point}\:\mathrm{and}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{remaining}\:\mathrm{ones}\:\mathrm{intersect}\:\mathrm{with}\:\mathrm{all} \\ $$$$\mathrm{other}\:\mathrm{lines}\:\mathrm{on}\:\mathrm{the}\:\mathrm{board}.\:\mathrm{Determine} \\ $$$$\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{intersections} \\ $$$$\mathrm{possible}. \\ $$

Question Number 111725 Answers: 1 Comments: 0

Find the positive integer n such that tan^(−1) ((1/3))+tan^(−1) ((1/4))+tan^(−1) ((1/5))+tan^(−1) ((1/n))=(π/4)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{5}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{n}}\right)=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 111724 Answers: 0 Comments: 6

How many real numbers x satisfy the equation 3^(2x+2) −3^(x+3) −3^x +3=0 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{3}^{\mathrm{2x}+\mathrm{2}} −\mathrm{3}^{\mathrm{x}+\mathrm{3}} −\mathrm{3}^{\mathrm{x}} +\mathrm{3}=\mathrm{0}\:? \\ $$

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