The least upper bound of a set in a poset is unique if it exists.
Let S be a poset and let A be a subset of S that has a least upper bound. We will show that this least upper bound is unique.
Suppose, for the sake of contradiction, that A has two distinct least upper bounds, denoted x and y. Since x is a least upper bound, we know that y is an upper bound of A and x is the least such upper bound. Similarly, we know that x is an upper bound of A and y is the least such upper bound.
Since x is an upper bound of A, we have y ≤ x, and since y is an upper bound of A, we have x ≤ y. This implies that x = y, which contradicts our assumption that x and y are distinct. Therefore, the least upper bound of a set in a poset is unique if it exists.
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What is the area, in square inches, of the trapezoid below?
3.6 in
8.5 in
5.1 in
5 in
The area of the given trapezoid is 33.48 in².
Given is a trapezoid, we need to find its area,
Area = sum of the parallel side × height/2
Therefore,
A = (1/2)(3.6)(5.1 + 8.5 + 5)
= 1.8(18.6)
= 33.48 sq. inches
Hence the area of the given trapezoid is 33.48 in².
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7 One Saturday afternoon, three friends decided to keep track of the number of text
messages they received each hour from 8 a.m. to noon. The results are shown below.
Emily said that the number of messages she received increased by 8 each hour.
Jessica said that the number of messages she received doubled every hour.
Chris said that he received 3 messages the first hour, 10 the second hour, none the third
hour, and 15 the last hour.
Which of the friends' responses best classifies the number of messages they received each
hour as a linear function?
1) Emily, only
2) Jessica, only
3) Emily and Chris
4) Jessica and Chris
A linear function is characterized by a constant rate of change.
1) Emily, only
Let's analyze each friend's claim:
Emily: She said her number of messages increased by 8 each hour.
This represents a constant increase and can be described as a linear function.
Jessica: She said her number of messages doubled every hour.
This represents an exponential growth and is not a linear function.
Chris: He received 3 messages the first hour, 10 the second hour, none the third hour, and 15 the last hour.
The changes are +7, -10, and +15, which are not constant.
This is not a linear function.
Based on this analysis, the answer is:
1) Emily, only.
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Find the value of x
52
(x+2)
2x
(x+10)
88°
Answer:
Step-by-step explanation:
7. Function g is represented by the graph. For what input value or values is g(x) = 4?
The given function g(x) is represented in the graph shown below.
It is clearly visible from the graph that g(x)=4 will be possible for only x=2 & x=-2.
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What unit of measurement is used to describe how far a set of values are from the mean? a) Variance b) Standard deviation c) Median d) Mode
The unit of measurement used to describe how far a set of values are from the mean is the standard deviation. Therefore, the correct answer is (b) standard deviation.
The variance is another measure of spread, but it is not in the form of the original units of measurement. The median is a measure of central tendency and not a measure of spread. The mode is the most frequently occurring value in a set and is also not a measure of spread.
The unit of measurement used to describe how far a set of values are from the mean is the Standard Deviation (b). It is calculated by taking the square root of the variance and provides a measure of the average distance between each value in the set and the mean. The Median (c), on the other hand, is the middle value in a set when the values are arranged in numerical order.
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Let X be a centered m x n-matrix. Then cov(X) = 2 XTX. Question 1 Not yet answered Points out of 10.00 Select one: O True Flag question O False
False, Let X be a centered m x n-matrix. Then cov(X) = 2 XTX. Because the correct formula for the covariance matrix is cov(X) = (1/(n-1)) XTX, where X is a centered m x n matrix and n is the number of observations (columns). The factor 2 in the given statement is incorrect, as the proper divisor should be (n-1) instead.
To answer your question, let X be a centered m x n matrix. Then cov(X) = 2 XTX. This statement is False.
The covariance matrix is a measure of the linear relationship between variables in a data set. For a centered matrix X, the covariance matrix cov(X) is defined as the matrix whose entries are given by:
cov(X) = (1/(n-1)) X^T X
where X^T is the transpose of X, and n is the number of observations in the data set (i.e., the number of columns in X).
The factor (1/(n-1)) is used instead of (1/n) to correct for bias when estimating the covariance matrix from a sample of data. This correction ensures that the estimated covariance matrix is an unbiased estimate of the true population covariance matrix.
The factor 2 in the given statement is incorrect. The expression 2 X^T X would lead to an overestimation of the covariance matrix, which would result in biased estimates of the population parameters. Hence, it is important to use the correct formula (cov(X) = (1/(n-1)) X^T X) when computing the covariance matrix for a centered matrix X.
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use double integrals to find the area inside the curve r = 3 + sin(θ).
The area inside the curve r = 3 + sin(θ) is (5π)/2 square units.
Double integration is an important tool in calculus that allow us to calculate the area of irregular shapes in the Cartesian coordinate system. In particular, they are useful when we are dealing with shapes that are defined in polar coordinates.
To find the area inside this curve, we can use a double integral in polar coordinates. The general form of a double integral over a region R in the xy-plane is given by:
∬R f(x,y) dA
where dA represents the infinitesimal area element, and f(x,y) is the function that we want to integrate over the region R.
In polar coordinates, we can express dA as r dr dθ, where r is the distance from the origin to a point in the region R, and θ is the angle that this point makes with the positive x-axis. Using this expression, we can write the double integral in polar coordinates as:
∬R f(x,y) dA = ∫θ₁θ₂ ∫r₁r₂ f(r,θ) r dr dθ
where r₁ and r₂ are the minimum and maximum values of r over the region R, and θ₁ and θ₂ are the minimum and maximum values of θ.
To find the area inside the curve r = 3 + sin(θ), we can set f(r,θ) = 1, since we are interested in calculating the area and not some other function. The limits of integration can be determined by finding the values of r and θ that define the region enclosed by the curve.
To do this, we first note that the curve r = 3 + sin(θ) represents a cardioid, which is a type of curve that is symmetric about the x-axis. Therefore, we only need to consider the region in the first quadrant, where 0 ≤ θ ≤ π/2.
To find the limits of integration for r, we note that the curve intersects the x-axis when r = 0. Therefore, the minimum value of r is 0. The maximum value of r can be found by setting θ = π/2 and solving for r:
r = 3 + sin(π/2) = 4
Therefore, the limits of integration for r are r₁ = 0 and r₂ = 4.
The limits of integration for θ are simply θ₁ = 0 and θ₂ = π/2, since we are only considering the region in the first quadrant.
Putting it all together, we have:
Area = ∬R 1 dA
= ∫[tex]0^{\pi /2}[/tex] ∫0⁴ 1 r dr dθ
Evaluating this integral gives us:
Area = π(3² - 2²)/2 = (5π)/2
Therefore, the area inside the curve r = 3 + sin(θ) is (5π)/2 square units.
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Using double integrals, the area inside the curve r = 3 + sin(θ) is 0 units².
For the area inside the curve r = 3 + sin(θ), we can use a double integral in polar coordinates. The area can be expressed as:
A = ∬R r dr dθ
where R represents the region enclosed by the curve.
In this case, the curve r = 3 + sin(θ) represents a cardioid shape. To determine the limits of integration for r and θ, we need to find the bounds where the curve intersects.
To find the bounds for θ, we set the expression inside sin(θ) equal to zero:
3 + sin(θ) = 0
sin(θ) = -3
However, sin(θ) cannot be less than -1 or greater than 1. Therefore, there are no solutions for θ in this case.
Since there are no intersections, the region R is empty, and the area inside the curve r = 3 + sin(θ) is zero.
Hence, the area inside the curve r = 3 + sin(θ) is 0 units².
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greg wants to move out of his dormitory and into an apartment near his college. his parents agreed, on the condition that the rent is no more than 25% of the cost of dorm living. to get an idea of rent amounts for one-bedroom apartments, greg looks at listings in a local newspaper and on an internet site. which answer best describes the sample and population?
Out of a whole population, a sample is a small part which is considered for any survey.
The population in this case would be all the one-bedroom apartments available for rent near the college.
The sample would be the listings that Greg is looking at in the local newspaper and on the internet site.
Sample: listings in a local newspaper and on an Internet site
Population: all available apartments near the college
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5) Ms. Ford is 48 and Ms. Lincoln is 35. How many
years ago was Ms. Ford exactly twice as old as Ms.
Lincoln?
Answer: 22 years ago.
Step-by-step explanation: Ms Ford and Ms Lincoln have a age gap of 13 years (48-35) so the last time Ms Ford would be exactly twice as old as Mis Lincoln would be when they were 13 and 26. From there we would just subtract that from their current age (48-26 or 35-13) giving us 22 as our answer.
express the integral as a limit of riemann sums. do not evaluate the limit. (use the right endpoints of each subinterval as your sample points.) ∫ 8 4 x 1 x 3 d x ∫48x1 x3dx
Thus, the limit is the definition of the definite integral ∫8^4 x/(x^3) dx, expressed as a limit of Riemann sums with right endpoints.
To express the integral ∫8^4 x/(x^3) dx as a limit of Riemann sums, we first need to partition the interval [4,8] into n subintervals of equal width Δx = (8-4)/n.
Let xi be the right endpoint of the ith subinterval, so xi = 4 + iΔx.
Then, the Riemann sum with right endpoints is:
∑(i=1 to n) f(xi)Δx
where f(xi) is the value of the function x/(x^3) at xi.
Substituting xi = 4 + iΔx and f(xi) = xi/(xi^3), we get:
∑(i=1 to n) (4+iΔx)/(4+i^3Δx^3) Δx
This is the expression for the Riemann sum with right endpoints.
To express the integral as a limit of Riemann sums, we take the limit of this expression as n goes to infinity:
lim(n->∞) ∑(i=1 to n) (4+iΔx)/(4+i^3Δx^3) Δx
This limit is the definition of the definite integral ∫8^4 x/(x^3) dx, expressed as a limit of Riemann sums with right endpoints.
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Suppose 10 is a factor of a. B and 8 is a factor of b. C, where a, b, and c are integers. What is the largest number that must be a factor of a. B. C?
A. 10
B. 20
C. 40 D. 80
The largest number that must be a factor of abc is 10 * 2^3 = 80.
Since 10 is a factor of a and 8 is a factor of b, we can express a and b in terms of their prime factorizations as:
a = 2^x * 5^y * k
b = 2^3 * k'
where x, y, and k are integers, and k' is an integer that may or may not contain a factor of 5 or k.
To find the largest number that must be a factor of abc, we need to find the prime factorization of c. Since a and b do not share any factors other than 1, the prime factorization of c can include any factors of 2, 5, or other prime factors that are not present in a or b.
The largest number that must be a factor of abc is the product of the largest powers of all the prime factors that appear in a, b, and c. Since a already contains the largest power of 5, and b already contains the largest power of 2, we just need to determine the largest power of 2 that appears in c.
Therefore, the largest number that must be a factor of abc is:
10 * 2^3 = 80
Therefore, the correct answer is D) 80.
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10. A shopkeeper earns a profit of Rs 1 by selling one pen and earns a loss of 30
paise on sale of one pencil. In a particular month, he incurs a loss of Rs 5. In that
month, he sold 40 pens. How many pencils did he sell in that period?
x^2+8x-5=0
x^2 + 12x + 4 = 0
x^2 + 18x + 90 = 0
–2x^2 – 12x – 9 = 0
4x^2 + 8x – 9 = 0
solve by completing the square root.
In Exercises 1-3, describe the domain of the function
A domain is the set of all possible values that can be used as input for a function. In other words, it is the set of all numbers for which the function is defined.
For example, if we have a function that calculates the area of a circle given its radius, the domain of this function would be all positive real numbers, because a circle cannot have a negative radius or zero radius. Now, when we are asked to describe the domain of a function in exercises 1-3, we need to figure out what values of the independent variable (usually denoted by x) can be used as input for the function. We need to look for any restrictions on x that might cause the function to be undefined. These restrictions could be due to a variety of reasons, such as division by zero, taking the square root of a negative number, or simply having a certain range of values that make sense for the context of the problem.
In summary, describing the domain of a function involves identifying the set of all possible input values that make sense for the function to be applied. It is an important concept in mathematics and is used in many different areas, from calculus to statistics.
In Exercises 1-3, the domain of the function refers to the set of all possible input values (independent variable) for which the function is defined. To describe the domain, you would need to identify the valid range of inputs for the given function in each exercise. For example, if a function has a square root or a denominator, the domain would exclude values that result in undefined outputs (like negative values under the square root or a zero in the denominator). Once you determine the domain for each exercise, you can express it using interval notation or inequalities to represent the allowable input values for the function in question.
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determine whether the sequence is increasing, decreasing,or not monotonic. is the sequence bounded? an
As the input values (n) increase, the value of the sequence term a_n = (n^2 - 3) / (n + 1) alternates between positive and negative values, indicating that the sequence is not monotonic.
To see this, consider the first few terms of the sequence:
a_1 = -2
a_2 = 1
a_3 = -4/2 = -2
a_4 = 7/3
As we can see, the sequence does not consistently increase or decrease as n increases.
To determine if the sequence is bounded, we can look at the limit of the sequence as n approaches infinity. We have:
lim (n -> infinity) (n^2 - 3) / (n + 1)
= lim (n -> infinity) (1 - 3/n^2) / (1 + 1/n)
= 1/1 = 1
Since the limit exists and is finite, the sequence is bounded. Specifically, we have:
-2 <= a_n <= 1
for all n.
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suppose the probability of success on each of 6 independent trials of an experiment is 0.2. (round your answers to four decimal places.)(a) what is the probability of 4 successes?
The probability of getting 4 successes out of 6 trials, where the probability of success on each trial is 0.2, is 0.0881 or approximately 0.0881.
To calculate the probability of 4 successes out of 6 trials, where the probability of success on each trial is 0.2, we can use the binomial probability formula.
The formula is P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where P(X=k) is the probability of getting k successes, n is the number of trials, p is the probability of success on each trial, and (n choose k) is the number of ways to choose k successes out of n trials.
Substituting the given values, we get:
P(X=4) = (6 choose 4) * (0.2)^4 * (0.8)^2
We can calculate the number of ways to choose 4 successes out of 6 trials using the combination formula, which is:
(6 choose 4) = 6! / (4! * 2!) = 15
Substituting this value into the formula, we get:
P(X=4) = 15 * (0.2)^4 * (0.8)^2
P(X=4) = 0.0881
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Suppose a population parameter is 0.8, and many large samples are taken
from the population. If the sample proportions are normally distributed, with
95% of the sample proportions falling between 0.704 and 0.896, what is the
standard deviation of the sample proportions?
A. 0.058
B. 0.078
C. 0.068
D. 0.048
The slope of the line that contains the points (3,5) and (3,6), is undefined
Yes, that is correct. The slope of the line that contains the points (3,5) and (3,6) is undefined.
To find the slope of a line, we use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line. In this case, both points have the same x-coordinate, which means that the denominator in the slope formula is zero. Division by zero is undefined, so the slope of the line is also undefined. Visually, this means that the line is vertical and does not have a defined slope. It is important to note that while the slope may be undefined, we can still determine other properties of the line, such as its x-intercept or y-intercept, and we can still graph the line using its equation or other methods.
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The length of the diagonal of rectangle sqrt(181) inches
The lengths of the sides a and b must satisfy the equation a^2 + b^2 = 181.
If we know that the length of the diagonal of a rectangle is sqrt(181) inches, we can use the Pythagorean theorem to find the length of the sides of the rectangle.
Let a and b be the lengths of the sides of the rectangle.
According to the Pythagorean theorem, the diagonal d is given by:
d^2 = a^2 + b^2
We are given that d = sqrt(181), so we can substitute that in and solve for one of the other variables:
(sqrt(181))^2 = a^2 + b^2
181 = a^2 + b^2
We can't determine the lengths of the sides without additional information, but we can say that the lengths of the sides a and b must satisfy the equation a^2 + b^2 = 181.
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What is the length of the sides of the rectangle if the length of the diagonal is sqrt(181) inches?
roll a fair die twice. suppose a and b are two independent events such that a=rolling a 3 in the first time and b=rolling a 3 in the second time. find p (a ∩ b).
the probability of both events A and B occurring, represented as P(A ∩ B), can be found by multiplying the individual probabilities of each event.
In this case, the probability of rolling a 3 on the first roll is 1/6, and the probability of rolling a 3 on the second roll is also 1/6, since the rolls are independent of each other. Therefore, the probability of both events occurring (rolling a 3 on both rolls) is:
P(A ∩ B) = P(A) x P(B)
P(A ∩ B) = (1/6) x (1/6)
P(A ∩ B) = 1/36
So the probability of rolling a 3 on both rolls is 1/36.
In explanation, when events are independent, the probability of both events occurring is found by multiplying their individual probabilities. This is because the outcome of one event does not affect the outcome of the other event.
In conclusion, the probability of rolling a 3 on both rolls is 1/36 when rolling a fair die twice, with A being rolling a 3 on the first roll and B being rolling a 3 on the second roll.
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What is the total discount percent as compared to the original price, if trousers are discounted by 12% and then a further 20% discount is given and new price is $281. 6
The total discount percent as compared to the original price is 29.6%.
Let P be the original price of the trousers.
According to the given information, the trousers are discounted by 12%, which means the new price becomes:
P' = P - 0.12P = 0.88P
After the first discount of 12%, a further 20% discount is given, which means the new price becomes:
P'' = P'(1 - 0.20) = 0.88P(0.80) = 0.704P
We are also given that the new price is $281.6, so we can set up an equation and solve for P:
0.704P = 281.6
P = 400
Therefore, the original price of the trousers is $400.
The total discount percentage is calculated as the difference between the original price and the final price, divided by the original price, multiplied by 100:
Discount percentage = ((P - P'')/P) x 100%
Substituting the values of P and P'', we get:
Discount percentage = ((400 - 281.6)/400) x 100%
Discount percentage = (118.4/400) x 100%
Discount percentage = 29.6%
Therefore, the total discount percent as compared to the original price is 29.6%.
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with 400 guests and breakfast costs of $550, what would be a hotel's food cost per guest?
Answer:
$1.38/guest
Step-by-step explanation:
$550/400 guests = 1.375 ≈ $1.38/guest
The hotel's food cost per guest for this breakfast event is $1.375.
Calculate the hotel's food cost per guest for a breakfast event with 400 guests and a total breakfast cost of $550.
Step 1: Identify the given values.
Total number of guests = 400
Total breakfast cost = $550
Step 2: Calculate the food cost per guest.
To find the food cost per guest, you'll need to divide the total breakfast cost by the total number of guests:
Food cost per guest = total breakfast cost / total number of guests
Food cost per guest = $550 / 400
Step 3: Compute the result
Food cost per guest = $1.375
So, the hotel's food cost per guest for this breakfast event is $1.375.
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Find the LCM of 18 and 20
Find the GCF of 105 and 135
The greatest factor they have in common is 15. LCM of 18 and 20:
We can find the LCM (Least Common Multiple) of 18 and 20 by listing their multiples until we find the first one that they have in common:
Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ...
Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, ...
So the LCM of 18 and 20 is 180.
GCF of 105 and 135:
We can find the GCF (Greatest Common Factor) of 105 and 135 by listing their factors and finding the greatest one they have in common:
Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105
Factors of 135: 1, 3, 5, 9, 15, 27, 45, 135
The greatest factor they have in common is 15. Therefore, the GCF of 105 and 135 is 15.
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braw a number line from -36 to 12
Answer:
Do youean draw or braw it's confusing
Use the expression 3x^2 + 4y - 5x + y + 11 to match the vocabulary terms with their "parts" of the given expression. constant, greatest common factor, number of terms in the polynomial, coefficient of the leading term, exponent of the leading term. DRAG & DROP THE ANSWER 4 11 1 3 2 6 5
The parts of the expression are:
Constant: 11
Greatest common factor: 1
Number of terms in the polynomial: 3
Coefficient of the leading term: -5
The exponent of the leading term: 2
We have,
Constant:
A constant is a term in an expression that does not have a variable attached to it. In this polynomial, the constant is 11.
Greatest common factor:
The greatest common factor (GCF) of a polynomial is the largest factor that all of its terms have in common.
In this case, the GCF is 1 (there are no common factors between the terms).
Number of terms in the polynomial: The number of terms in a polynomial is simply the count of all the individual terms. In this case, there are three terms: 3x², 4y, and -5x + y + 11.
Coefficient of the leading term:
The coefficient of a term is the number that is multiplied by the variable. In the leading term, which is the term with the highest degree (highest exponent), the coefficient is 3. So the coefficient of the leading term in this polynomial is 3.
The exponent of the leading term:
The exponent of the leading term is simply the degree of the polynomial, which is the highest exponent of any term.
In this polynomial, the highest degree is 2 (from the term 3x²), so the exponent of the leading term is 2.
Thus,
The parts of the expression are:
Constant: 11
Greatest common factor: 1 (since there are no common factors that can be factored out)
Number of terms in the polynomial: 3
Coefficient of the leading term: -5
The exponent of the leading term: 2
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4. a study of peach trees found that the average number of peaches per tree was 625. the standard deviation of the population is 35 peaches per tree. a scientist wishes to find the 95% confidence interval for the mean number of peaches per tree. how many trees does she need to sample to obtain an average accurate to within 20 peaches per tree? round to the nearest whole number. (10 point
The scientist needs to sample at least 52 trees to obtain an average accurate to within 20 peaches per tree.
The formula for the sample size required to estimate the population mean with a specified margin of error is:
n = (z² * σ²) / E²
where:
n is the required sample size
z is the z-score for the desired level of confidence (in this case, 1.96 for 95% confidence)
σ is the population standard deviation (35 peaches per tree)
E is the desired margin of error (20 peaches per tree)
Plugging in the given values, we get:
n = (1.96² * 35²) / 20²
n ≈ 52
Therefore, the scientist needs to sample at least 52 trees to obtain an average accurate to within 20 peaches per tree, with 95% confidence.
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an opt frame and a true/false condition on a message serve essentially the same purpose.
T/F
True, an optional frame (opt frame) and a true/false condition on a message do serve essentially the same purpose in the context of communication protocols and data representation.
Both of these elements are used to control the structure and flow of data by adding conditional statements that determine whether certain parts of the data should be included or excluded based on specific conditions.
An opt frame is a data structure used in protocol design that allows for the optional inclusion of information. It is a container for data that may or may not be present in the final message, depending on the circumstances or requirements. The inclusion of an opt frame is determined by specific conditions and can be used to customize the message structure, enabling flexibility in communication.
Similarly, a true/false condition on a message is used to specify whether certain elements should be included or excluded based on the evaluation of a specific condition. This mechanism enables messages to be customized according to certain criteria, allowing the sender and receiver to communicate more effectively by focusing only on the necessary information.
Both opt frames and true/false conditions serve the purpose of providing flexible data structures and improving the efficiency of communication protocols. They allow for the customization of message structures based on specific conditions, ensuring that only relevant information is transmitted between parties.
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person who replies with an answer gets 12 points
The difference in the amount of interest that Holly would have to pay for each of the loans is: $1400
How to find the simple interest?The formula for simple interest is:
Interest = Principal * Rate * Time/100
For the first one, we have:
Principal = $10000
Rate = 4%
Time = 4 years
Thus:
Interest = 10,000 * 4/100 * 4 years
Interest = $1,600 Interest Holly has to pay on 4%, 4-year loan.
For the second one, we have:
Interest = 10,000 * 5/100 * 6 years
Interest = $3,000 Interest Holly has to pay on 5%, 6-year loan.
$3,000 - $1,600 =$1,400 Difference in interest between the two loan choices.
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A soma de dez termos consecutivos da sucessão 2,3,5,8,11 é
Primeiro, vamos identificar a razão da sequência. A diferença entre cada termo consecutivo é:
3 - 2 = 1
5 - 3 = 2
8 - 5 = 3
11 - 8 = 3
Podemos ver que a razão é 3.
Agora, podemos usar a fórmula para a soma dos n primeiros termos de uma progressão aritmética:
Sn = (n/2)(a1 + an)
onde Sn é a soma dos n primeiros termos, a1 é o primeiro termo e an é o último termo.
Para encontrar a soma dos próximos dez termos da sequência, precisamos primeiro encontrar o valor do sexto termo. Podemos fazer isso adicionando a razão ao quinto termo:
11 + 3 = 14
Agora, podemos usar a fórmula acima para encontrar a soma dos próximos dez termos:
S10 = (10/2)(14 + 3(10))
S10 = 5(44)
S10 = 220
Portanto, a soma dos próximos dez termos é 220.
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Find the sum of the infinite geometric sequence that begins 1/12, 1/18, 1/27
The sum of the infinite geometric sequence that begins with 1/12, 1/18, 1/27 is 1/4.
To find the sum of the infinite geometric sequence that begins with 1/12, 1/18, 1/27, we first need to determine the common ratio, r.
To find the common ratio, we divide each term by the preceding term, as follows:
r = (1/18) ÷ (1/12) = 1/18 × 12/1
= 2/3
r = (1/27) ÷ (1/18) = 1/27 × 18/1
= 2/3
Since the common ratio is the same for all terms, this is a geometric sequence.
The formula for the sum of an infinite geometric sequence is:
sum = a / (1 - r)
where a is the first term and r is the common ratio.
Substituting the values we found, we get:
sum = (1/12) / (1 - 2/3)
= (1/12) / (1/3)
= 1/4
Therefore, the sum of the infinite geometric sequence that begins with 1/12, 1/18, 1/27 is 1/4.
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