Q6. The Binomial Theorem says that given any a, b e R and an integer n > 0 п n (α + β)Σ (a + b)" = 3 C) . Jan-kek k k=0 Prove the formula and use it to prove the identities: = 0 n 1. k= (%) = 21 (
Answers
These identities are derived from the Binomial Theorem and hold for all positive integers n.
The Binomial Theorem states that for any real numbers a and b and a positive integer n, the expansion of (a + b)^n can be expressed as:
(a + b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + C(n,2) * a^(n-2) * b^2 + ... + C(n,n-1) * a^1 * b^(n-1) + C(n,n) * a^0 * b^n
Where C(n,k) represents the binomial coefficient, given by the formula:
C(n,k) = n! / (k! * (n-k)!)
Now, let's prove the formula and use it to prove the given identities.
Proof of the Binomial Theorem:
To prove the formula, we can use mathematical induction. The base case (n = 1) is straightforward:
(a + b)^1 = a + b
Assuming the formula holds for some integer k, we need to prove it holds for k + 1:
(a + b)^(k+1) = (a + b) * (a + b)^k
Expanding the right-hand side using the induction hypothesis, we get:
(a + b)^(k+1) = (a + b) * [C(k,0) * a^k * b^0 + C(k,1) * a^(k-1) * b^1 + ... + C(k,k-1) * a^1 * b^(k-1) + C(k,k) * a^0 * b^k]
Distributing and simplifying, we obtain:
(a + b)^(k+1) = C(k,0) * a^(k+1) * b^0 + C(k,1) * a^k * b^1 + ... + C(k,k-1) * a^2 * b^(k-1) + C(k,k) * a^1 * b^k + C(k,0) * a^0 * b^(k+1)
By comparing this expression with the original expansion of (a + b)^(k+1), we see that it matches the form of the Binomial Theorem. Therefore, the formula holds for all positive integers n.
Using the Binomial Theorem, we can now prove the given identities:
1. (1 + x)^n = C(n,0) * 1^n * x^0 + C(n,1) * 1^(n-1) * x^1 + ... + C(n,n-1) * 1^1 * x^(n-1) + C(n,n) * 1^0 * x^n
Simplifying the coefficients, we get:
(1 + x)^n = C(n,0) + C(n,1) * x + C(n,2) * x^2 + ... + C(n,n-1) * x^(n-1) + C(n,n) * x^n
Since C(n,k) = C(n,n-k), the terms with odd powers of x will cancel out when we add them to the corresponding terms with even powers of x. Therefore, the sum of the odd-powered terms is equal to the sum of the even-powered terms, which proves the identity.
2. (1 - x)^n = C(n,0) * 1^n * (-x)^0 + C(n,1) * 1^(n-1) * (-x)^1 + ... + C(n,n-1) * 1^1 * (-x)
^(n-1) + C(n,n) * 1^0 * (-x)^n
Simplifying the coefficients and factoring out (-1)^k, we get:
(1 - x)^n = C(n,0) - C(n,1) * x + C(n,2) * x^2 - ... + (-1)^k * C(n,k) * x^k + ... + (-1)^n * C(n,n) * x^n
Again, by observing the cancelation of terms with odd powers of x, we can see that the sum of the odd-powered terms is equal to the sum of the even-powered terms, proving the identity.
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Related Questions
What are the two zeroes/solutions for the following equation?
(x - 5)(x + 2) = 0
X =
and
Hellpppppo MEEEEEEE
Answers
X=5
Step-by-step explanation:
X-5=0. X+2=0
X=5. X=-2
#21 please I need help
Answers
Answer:
12y/5
Step-by-step explanation:
Justin notices a particular type of caterpillar feeds only on cottonwood trees in his neighborhood.In which way has Justin increased his scientific knowledge?
He has made observations about the natural world.
He has used technology.
He has performed a practical experiment.
He has performed an unethical experiment.
Answers
Justin has increased his scientific knowledge by making observations about the natural world.
Justin's act of noticing a particular type of caterpillar feeding only on cottonwood trees in his neighborhood is an example of making observations about the natural world. Observations are a fundamental part of the scientific process and play a crucial role in increasing scientific knowledge.Observations involve using our senses to gather information and data about the world around us. By carefully observing the behavior of the caterpillars and noting their specific feeding habits on cottonwood trees, Justin is gathering valuable information about the natural phenomenon. This firsthand observation provides him with direct evidence and insights into the ecological relationship between the caterpillars and the cottonwood trees.Observations are the foundation of scientific inquiry, as they provide the basis for asking questions, formulating hypotheses, and conducting further investigations. Justin's act of observing the caterpillars' feeding behavior on cottonwood trees contributes to our understanding of the specific dietary preferences and ecological interactions of these caterpillars.In contrast, the options of using technology, performing a practical experiment, or performing an unethical experiment do not accurately describe Justin's actions. While technology and experiments can be valuable tools in scientific inquiry, the given scenario does not mention their usage. Furthermore, the mention of an unethical experiment is not applicable or supported by the information provided.Therefore, Justin has increased his scientific knowledge by making observations about the natural world, specifically regarding the feeding behavior of the caterpillars on cottonwood trees.For more such questions on Justin, click on:
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dairy queen is a take-out yogurt shop owned by linda smith. customers arrive at a rate of 25 per hour. linda serves a customer, on the average, in 1.5 minutes. assume that the arrivals and the service times are poisson distributed and exponentially distributed, respectively. what is the probability that a customer will wait in a line between 3 to 6 minutes?
Answers
The probability that a customer will wait in a line between 3 to 6 minutes is about 16.58%.
To solve this problem, we need to use the Poisson process and the exponential distribution.
Let X be the number of arrivals in a 3-minute interval. Since customers arrive at a rate of 25 per hour, the expected number of arrivals in a 3-minute interval is:
λ = (25/60) x 3 = 1.25
Thus, X is Poisson distributed with parameter λ = 1.25.
Let Y be the service time for a customer. Since Linda serves a customer, on average, in 1.5 minutes, Y is exponentially distributed with parameter μ = 1/1.5 = 0.6667.
Let Z be the waiting time for a customer in the line. Z is the sum of X independent service times, so Z is gamma distributed with parameters X and μ.
We want to find the probability that Z is between 3 and 6 minutes:
P(3 ≤ Z ≤ 6) = ∫∫ f(x,y) dx dy
where f(x,y) is the joint probability density function of X and Y:
f(x,y) = (λ^x / x!) e^(-λ) μ e^(-μy) = (1.25^x / x!) e^(-1.9167) e^(-0.6667y)
Now we can evaluate the double integral:
P(3 ≤ Z ≤ 6) = ∫∫ f(x,y) dx dy
= ∫∫ (1.25^x / x!) e^(-1.9167) e^(-0.6667y) dx dy
= ∫ e^(-0.6667y) e^(-1.9167) ∑ (1.25^x / x!) dx dy (x=0 to ∞)
= e^(-1.9167) ∫ e^(-0.6667y) e^(1.25) dy ∑ (1.25^x / x!) (x=0 to ∞)
The sum in the integral is the Taylor series expansion of e^1.25, which is equal to e^1.25 = 3.4903.
Using a table of integrals or a computer software, we can evaluate the integral:
∫ e^(-0.6667y) e^(1.25) dy = (1/0.6667) (e^(1.25) - e^(-0.6667(6))) = 3.7225
Therefore, the probability that a customer will wait in a line between 3 to 6 minutes is:
P(3 ≤ Z ≤ 6) = e^(-1.9167) ∫ e^(-0.6667y) e^(1.25) dy ∑ (1.25^x / x!) (x=0 to ∞)
= e^(-1.9167) (3.7225) (3.4903) = 0.1658 or about 16.58% (rounded to four decimal places).
Therefore, the probability that a customer will wait in a line between 3 to 6 minutes is about 16.58%.
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Deion has the following data:
Z 15 20 14 9 13 19 10
If the mode is 20, which number could z be? The answer choice is
10
20
Answers
Answer:
20
Step-by-step explanation:
The mode of a set is the most frequent data value. In this case, for 20 to be the mode, Z would have to be 20 as well. This way, 20 appears most out of all the other data values.
Answer:
20
Step-by-step explanation:
Total surface area??
Answers
Answer:
150 sq.cm
Step-by-step explanation:
Surface area = (5 cm x 5 cm) x 6 faces
Surface area = 25 sq.cm x 6 faces
Surface area = 150 sq.cm
Hope that helps!
Answer:
150
Step-by-step explanation:
p(x)=1/2(x-2)(2x-3)(4x+7)
Answers
The simplified expression for the polynomial is 2y = 8x³− 14x²− 25x+ 42
A mathematical formula that uses integer variables, constants, and algebraic operations is known as an algebraic expression (addition, subtraction, multiplication, division, and exponentiation by a rational exponent).
A polynomial is an equation made up of exponents, constants, and variables that is combined using mathematical operations including addition, subtraction, multiplication, and division (No division operation by a variable)The given function is p(x)=1/2(x-2)(2x-3)(4x+7)
The simplified form of the function can be written as
2y = 8x³− 14x²− 25x+ 42
Now we know that the roots of the equation are 2 ,3/2 and -7/4.
The graph of the function is attached below.
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You start at (9, 10). You move down 6 units. Where do you end?
Answers
Answer:
(9,4)
Step-by-step explanation:
(x,y)
X is horizontal, Y is vertical.
10-6=4
Answer:
3,10
Step-by-step explanation:
going down would decrease the y axis so lowering the first number
Create a list of steps, in order, that will solve the following equation. (x-5)^2=25(x−5) 2 =25left parenthesis, x, minus, 5, right parenthesis, squared, equals, 25 Solution steps:
Answers
The value of x in the equation (x - 5)² = 25 is 10.
How to solve an equation?(x - 5)² = 25
Take the square root of both sides
Therefore,
√(x - 5)² = √25
x - 5 = 5
add 5 to both sides
x - 5 + 5 = 5 + 5
x = 10
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Question 4 (0.5 points)
Which expression is equivalent to 42x + 6?
6(7x + 6)
0617x + 1)
7(6x + 6)
2(40x + 8)
Answers
While shopping for a big party Nick has $90 worth of item) in his grocery cart. Soda is $2 a
bottle. Nick can spend $112 on everything. How many bottles of soda can Nick buy?
Answers
one of the questions rasmussen reports included on a 2018 survey of 2,500 likely voters asked if the country is headed in right direction. representative data are shown in the datafile named rightdirection. a response of yes indicates that the respondent does think the country is headed in the right direction. a response of no indicates that the respondent does not think the country is headed in the right direction. respondents may also give a response of not sure. (a) what is the point estimate of the proportion of the population of respondents who do think that the country is headed in the right direction? (round your answer to four decimal places.) (b) at 95% confidence, what is the margin of error for the proportion of respondents who do think that the country is headed in the right direction? (round your answer to four decimal places.) (c) what is the 95% confidence interval for the proportion of respondents who do think that the country is headed in the right direction? (round your answers to four decimal places.) to (d) what is the 95% confidence interval for the proportion of respondents who do not think that the country is headed in the right direction? (round your answers to four decimal places.) to (e) which of the confidence intervals in parts (c) and (d) has the smaller margin of error? why? the confidence interval in part (c) has a ---select--- margin of error than the confidence interval in part (d). this is because the sample proportion of respondents who do think that the country is headed in the right direction is ---select--- than the sample proportion of respondents who do not think that the country is headed in the right direction.
Answers
The confidence interval with the smaller margin of error will be the one with the smaller sample proportion difference.
(a) To calculate the point estimate of the proportion of respondents who think the country is headed in the right direction, divide the number of "yes" responses by the total number of respondents (2,500).
Point estimate = (number of "yes" responses) / 2,500
(b) To find the margin of error at 95% confidence, use the formula:
Margin of error = Z * √(p(1-p) / n)
Here, Z = 1.96 (from the standard normal distribution for 95% confidence), p is the point estimate calculated in part (a), and n = 2,500.
(c) To find the 95% confidence interval for the proportion of respondents who think the country is headed in the right direction, use the formula:
Confidence interval = point estimate ± margin of error
(d) To find the 95% confidence interval for the proportion of respondents who do not think the country is headed in the right direction, first calculate the point estimate for "no" responses:
Point estimate (no) = (number of "no" responses) / 2,500
Then calculate the margin of error and confidence interval as in parts (b) and (c).
(e) To determine which confidence interval has a smaller margin of error, compare the margin of error calculated in parts (b) and (d).
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a bag contains 4 white 5 red and 6 blue balls three balls are drawn at radon from the bag the probality that all of them are red is
Answers
The probability that all three balls drawn from the bag are red is 6/273.
What is probability?Prοbability is a measure οf the likelihοοd οr chance that a particular event will οccur. It quantifies the uncertainty assοciated with an οutcοme in a given situatiοn οr experiment.
Given:
- Total number of balls in the bag: 4 white + 5 red + 6 blue = 15 balls
- Number of red balls: 5
For the first draw, the probability of selecting a red ball is 5 red / 15 total balls = 1/3.
After the first red ball is drawn, there are 4 red balls left and 14 total balls remaining in the bag. Therefore, for the second draw, the probability of selecting another red ball is 4 red / 14 total balls = 2/7.
After the second red ball is drawn, there are 3 red balls left and 13 total balls remaining in the bag. Therefore, for the third draw, the probability of selecting the final red ball is 3 red / 13 total balls.
To find the probability of all three balls being red, we multiply the individual probabilities together:
P(all red) = (1/3) * (2/7) * (3/13)
Simplifying the expression, we get:
P(all red) = 6/273
Therefore, the probability that all three balls drawn from the bag are red is 6/273.
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What is the solution(s) to the equation 5a2 - 44 = 81?
A
+25
B
-5
55
D
1125
Answers
Answer:
B
Step-by-step explanation:
5a^2 - 44 = 81 Add 44 to both sides
5a^2 = 81 + 44 Collect like terms
5a^2 = 125 Divide both sides by 5
a^2 = 125/5
a^2 = 25 Take the square root of both sides
√a^2 = √25
a = +5
a = -5
The answer is B
20x50= ?
Please answer me!
Answers
Answer:
1,000
Step-by-step explanation:20*50
What is the midpoint of the line segment with the given endpoints?
(-12, 9) and (8, -21)
Enter your answer by filling in the boxes.
Answers
The line segment's midpoint is\((\frac{-5}{8}, \frac{-5}{6})\)
How can you find the middle of a line?
The distance between the two end points should be divided by two. This distance from either end is where the centre of that line is situated. You may also add the two endpoint x coordinates and divide the result by 2. the same is true for the y coordinates.
The midway is the point where a line segment splits into two equal parts. It is located at the middle of the line section, equally distant from both of its endpoints.
\(((\frac{(x1 + x2)}{2}, (\frac{(y1 + y2)}{2})\) is the midpoint.
In light of the posed query:
The midpoint of the line segment with endpoints is determined using this formula.
\((\frac{3}{8},\frac{ -2}{3} and \frac{-3}{4}, \frac{-1}{9} is:\)
\((\frac{(3/8 + -3/4)}{2}, \frac{(-2/3 + -1/9)}{2} = (\frac{-5}{8}, \frac{-5}{6})\)
The line segment's midpoint is therefore\((\frac{-5}{8}, \frac{-5}{6})\)
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PLEASE HELP GIVE TWO DIFFERENT DECIMALS THAT ROUND TO 5.24 WHEN ROUNDING TO THE NEAREST HUNDREDTHS
Answers
Answer:
5 and 3
Step-by-step explanation:
5.235 rounded off to the nearest hundredths is 5.23|5->5.24
5.233 rounded off to the nearest hundredths is 5.23|3->5.24
For an arithmetic sequence a,= 7+3(n-1), what is the 31st term?a.) 97b.) 100c.) 91d.) 94
Answers
For an arithmetic sequence a,= 7+3(n-1)
when n is 31
For an arithmetic sequence a,= 7+3(31-1)
An arithmetic sequence a,= 7+3(30)
An arithmetic sequence a,= 7+90
An arithmetic sequence a= 97
THE CORRESCT
A line passes through the points ( 1 , 2 ) and ( 5 , 3 ) . Another line passes through the points ( 5 , 3 ) and ( 0 , 0 ) . Will the two lines intersect
Answers
Yes these two lines intersect with each other.
Given,
Co ordinates of line 1 : ( 1 , 2 ) , ( 5 , 3 )
Co ordinates of line 2 : ( 5 , 3 ) , ( 0 , 0 )
So,
Intersection is the meeting point of any lines at a common co ordinate.
The first line has the co ordinates of (1,2) and (5,3).
Similarly second line also has the co ordinate (5,3) and passes through origin.
Since both the lines has one point in common, that means that two lines intersect with each other at point (5,3).
Thus the two given lines will intersect each other at point (5,3)
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What is the critical value z* for constructing an 80% confidence interval?
Answers
Answer:khan is 1.282 or c
Step-by-step explanation:
12
Calculate the area of the given segment. Round your answer to the nearest tenth, if necessary.
60
8 in.
Answers
Check the picture below.
\(\textit{area of a segment of a circle}\\\\ A=\cfrac{r^2}{2}\left( ~~ \cfrac{\pi \theta }{180}-\sin(\theta ) ~~ \right) \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=8\\ \theta =60 \end{cases} \\\\\\ A=\cfrac{8^2}{2}\left( ~~ \cfrac{\pi (60) }{180}-\sin(60^o ) ~~ \right)\implies A=32\left( ~~ \cfrac{\pi }{3}-\sin(60^o ) ~~ \right) \\\\\\ A=32\left( ~~ \cfrac{\pi }{3}-\cfrac{\sqrt{3}}{2} ~~ \right)\implies A=\cfrac{32\pi }{3}-16\sqrt{3}\implies A\approx 5.8~in^2\)
an equation of a number and 25 equals to the sum of 141 and 159 divided by 2
Answers
Answer:
25 = 141 + 159/2
Step-by-step explanation:
Please help thank you
Answers
The value of [x] in the figure given is 54.
What are Alternate interior angles?
Alternate interior angles are the angles formed when a transversal intersects two coplanar lines. They lie on the inner side of the parallel lines but on the opposite sides of the transversal.
Given are two parallel lines.
The two lines are parallel and are intersected by a transversal. Assume
that -
∠1 = (x +3)°
∠2 = (2x - 61)°
Now, the angle at the vertically opposite position with respect to ∠2 will have same measurements. Let's call this ∠3 = (2x - 61)°.
Now, ∠3 and ∠1 are alternate interior angles and their measures will be equal. So, we can write -
∠1 = ∠3
x + 3 = 2x - 61
3 + 61 = 2x - x
x = 64
The value of [x] is 54
Therefore, the value of [x] in the figure given is 54.
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your friend is binging half of a crate of popsicles to the picnic. of you put what is left in the crate evenly into 6 ice buckets, what fraction of the crate will go into each of ice bucket
Answers
Thus, the fraction of crate that will go in each of the ice bucket is found as - 1/12.
Explain about the fraction:An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction.
Fractions with a numerator less than the denominator are said to be proper fractions. When the numerator exceeds the denominator, the fraction is said to be inappropriate.
Given data:
fraction of crate brought by friend = 1/2
Total number of ice buckets = 6
Fraction of crate in each bucket = fraction of crate brought by friend / Total number of ice buckets
Fraction of crate in each bucket = (1/2) / 6
Fraction of crate in each bucket = 1 / (2*6)
Fraction of crate in each bucket = 1/12
Thus, the fraction of crate that will go in each of the ice bucket is found as - 1/12.
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Complete the following similarity statement for the figure below.
Answers
Answer:
Triangle JKLM ~ Triangle STUV
i keep getting the wrong answer and have no idea what to do
Answers
Answer:
[C] 0
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightDistributive Property
Algebra I
Terms/CoefficientsFunctionsFunction NotationCalculus
Limits
Derivatives
Definition of a Derivative: \(\displaystyle f'(x)= \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle\displaystyle g(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)
\(\displaystyle f(x) = \frac{3}{5x^4 + 3}\)
\(\displaystyle g(0)\)
Step 2: Differentiate
Substitute in x [Function g(x)]: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}\)Substitute in function f(x) [Function g(x)]: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{\frac{3}{5(0 + h)^4 + 3} - \frac{3}{5(0)^4 + 3}}{h}\)Simplify: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{\frac{3}{5h^4 + 3} - 1}{h}\)Rewrite: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{3}{h(5h^4 + 3)} - \frac{1}{h}\)Rewrite: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{3}{h(5h^4 + 3)} - \frac{5h^4 + 3}{h(5h^4 + 3)}\)[Subtraction] Combine like terms: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{3 - 5h^4 + 3}{h(5h^4 + 3)}\)[Addition] Simplify: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{-5h^4 + 6}{h(5h^4 + 3)}\)[Distributive Property] Distribute h: \(\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{-5h^4 + 6}{5h^5 + 3h}\)Evaluate limit [Power Method]: \(\displaystyle\displaystyle g(0) = 0\)Since the bottom polynomial has a higher degree than the top polynomial, the bottom polynomial will increase faster.
∴ If the bottom is approaching a bigger value, the fraction gets smaller and smaller, approaching 0.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
Mindy is the head cheerleader on her suad. She is making banners to place throughout the school for the first football game of the season. She used ⅔ cup of glitter to make 1 cup of glittery paint. Using this same proportion, how many cups of glitter will Mindy use to make 9 cups of glittery paint?
Answers
Answer:
\(6\:\text{cups}\)
Step-by-step explanation:
Since the proportion is maintained, we can set up the following proportion:
\(\frac{2/3}{1}=\frac{x}{9}\), where \(x\) is the amount of glitter, in cups, Mindy will use to make 9 cups of glittery paint.
Cross-multiply to get:
\(1x=9\cdot \frac{2}{3},\\x=9\cdot \frac{2}{3}=\boxed{6\:\text{cups}}\)
The number of times an item occurs in a data set is called what?
Answers
Answer:
Frequency
Step-by-step explanation:
Hope this helps.
The sum of 3 times a number and 6 is 8.
Answers
Answer:
The number is 4/9
Step-by-step explanation:
3(6*(4/9))=3*(2+(2/3))=8
5 pts) Let outer (exterior) measure of En CR satisfy m+ (En) ≤ 2¯n, n = 1, 2,.... Prove that (4 pts) M* (ů) 1.) U En ≤1. n=1 Give an example of En such that in the latter formula the equality holds. (1 pt)
Answers
The given condition states that for each positive integer n, the outer measure of En, denoted as m+(En), is less than or equal to 2 raised to the power of minus n.
To prove the given statement, let's consider the countable collection of sets {En} with indices ranging from 1 to infinity. By the definition of outer measure, we have m+(En) ≤ 2^(-n) for each n.
Now, we can use the subadditivity property of outer measure, which states that for any collection of sets, the measure of their union is less than or equal to the sum of their individual measures. Applying this property to the collection {En}, we have:
m+(∪En) ≤ Σ(m+(En)) (where Σ denotes summation)
Since m+(En) ≤ 2^(-n) for each n, we can rewrite the above inequality as:
m+(∪En) ≤ Σ(2^(-n))
This is a geometric series with a common ratio of 1/2. The sum of this series is 1, which means:
m+(∪En) ≤ 1
Thus, we have proven that the measure of the union of all En is less than or equal to 1.
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