Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
7 + 6 + 36/7 + 216/49 + ...
If it is convergent, find its sum.
Answers
Answer:
- L < 1, so by series ratio test, It is convergent
- sum of the convergent series is 49
Step-by-step explanation:
Given the data in the question;
series = 7 + 6 + 36/7 + 216/49 + .........
⇒ 7 + 7 × 6/7 + 7 × 36/49 + 7 × 216/343 +........
⇒ 7 + 7 × 6/7 + 7 × 6²/7² + 7 × 6³/7³ .....
⇒ 7( 1 + 6/7 + 6²/7² + 6³/7³ + ..... )
⇒ 7 ∞∑_\(_{n=0\) \((\) 6/7 \()^n\)
⇒ ∞∑_\(_{n=0\) ( \(6^n\)/\(7^{n-1\) )
So, L = \(\lim_{n \to \infty} | \frac{a_{n} + 1}{a_n} |\)
= \(\lim_{n \to \infty} | \frac{\frac{6^n+1}{7n} }{\frac{6^n}{7^n-1} } |\)
= \(\lim_{n \to \infty} | \frac{6}{7} |\)
= 6 /7
L < 1
so by series ratio test,
It is convergent
So we find the sun;
Sum of infinite geometric series is;
⇒ a / (1-r)
here, a = first number and r = common ratio
∑ = 7( 1 + 6/7 + 6²/7² + 6³/7³ + ..... )
a = 1 and r = 6/7
so
∑ = 7( \(\frac{1}{1 - \frac{6}{7} }\) )
= 7( \(\frac{1}{\frac{7-6}{7} }\) )
= 7( \(\frac{1}{\frac{1}{7} }\) )
= 7( 7 )
= 49
Therefore, sum of the convergent series is 49
Related Questions
Calculate the loss on selling 50 shares of stock originally bought at 13 3/4 and sold at 12
Answers
The loss on selling 50 shares of stock would be 87.50.
To calculate the loss on selling 50 shares of stock originally bought at
\(13\frac{3}{4}\) and sold at 12, we need to determine the difference between the purchase price and the selling price, and then multiply it by the number of shares sold.
First, let's convert the purchase price from a mixed fraction to a decimal. \(13\frac{3}{4}\) can be expressed as 13.75.
Next, we calculate the difference between the purchase price and the selling price:
\(13.75 - 12 = 1.75.\)
Finally, we multiply the difference by the number of shares sold:
\(1.75 \times50 = 87.50.\)
Therefore, the loss on selling 50 shares of stock would be 87.50.
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What is the value of the expression? Do not use a calculator.
tan
Tan2pie/3
Answers
The value of tan 2π/3 without calculator is -√3.
What is the value of tan 2π / 3 without calculator?The value of tan 2π/3 without calculator is calculated by applying trig identities as follows;
the value of π = 180 degrees
So we can replace the value of π in the function with 180 degrees as follows;
tan ( 2π / 3) = tan (2 x 180 / 3)
tan (2 x 180 / 3) = tan (2 x 60)
tan (2 x 60) = tan (120)
tan (120) if found in the second quadrant, and the value will be negative since only sine is positive in the second quadrant.
tan (120) = - tan (180 - 120)
= - tan (60)
= -√3
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What is the first step to solving the following equation?5x – 11 = 42
Answers
Answer:
add 11 on both sides
Step-by-step explanation:
to solve this, you want x alone on one side. To achieve this, you first add 11 on both sides, so you only have the 5x alone.
Second step then is something to get only one x on the left side ;-)
(divide both sides by 5)
Answer:
the first step is to get the x term by itself on one side
Which raises can be represented by algebraic expression 2x + 1 check all that apply
Answers
Answer:
two times a number plus 1
Step-by-step explanation:
PLEASE HELP!!!
ASAP :O
NO LINKS!!!
Answers
Answer:
D.
Step-by-step explanation:
The magnitude of a number is its absolute value.
|-5.8| = 5.8
Answer: D.
Jerry ran 4/3 of a mile in 4/18 of an hour. What was Jerry's rate of speed?
⚫️4 miles per hour
⚫️6 miles per hour
⚫️5_1/3 miles per hour
⚫️8/27 miles per hour
Answers
Answer:
B, 6 miles per hour
Step-by-step explanation:
1. 4/3 of a mile in 4/18 of an hour, rewrite as (4/3 / 4/18)
2. Dividing fractions is multiplying by the reciprocal, so (4/3 * 18/4)
3. (4/3 * 18/4 = 72/12)
4. 72/12 can be simplified to 6/1 or 6.
5. Answer is B, 6 miles per hour
mathmatically indication example with solution
Answers
A mathematically indication example would be to solve the equation 3x + 2 = 14 for the value of x. The solution would be 4.
How to solve the equation ?Looking for a mathematically indication example, we can consider a simple mathematical equation with one variable and solve it.
The equation would be 3 x + 2 = 14.
So we can solve for the equation to be :
3x + 2 - 2 = 14 - 2
3x = 12
3 x / 3 = 12 / 3
x = 4
In conclusion, the mathematically indication example would be 3x + 2 = 14 and the value of x would be 4.
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Which of the following is a solution for x in the equation 5x + 3 = 33?
Answers
Answer:
x=6
Step-by-step explanation:
Answer: x=6
Step-by-step explanation: 5•6+3=33
Good luck! :)
A car uses 8 gallons of gasoline to travel 192 miles. How many gallons are used to go 144 miles? *
Answers
Answer:
6 gallons
Step-by-step explanation:
You can find this using the ration method.
If, 8 gallons : 192 miles
x gallons : 144 miles
We must find the value of x, which is the number of gallons used to travel 144 miles.
Cross multiply the ratios.
192 x = 8 × 144
192 x = 1152
x = 1152 ÷ 192
x = 6
∴ It takes 6 gallons to travel 144 miles.
Jessica makes $20 per hour. She earns a raise of 15%. How much will her new salary per hour be?
Answers
Answer:
$23 a hour
Step-by-step explanation:
yossi used to have a square garage with 364ft of floor space. recently built an addition to it. The garage is still a square, but now it has 50% more floor space. What was the length of one side of the garage originally? What is the length of one side of the garage now? What was the percent increase in the length of one side?
Answers
The original length of the square garage is 19.1 ft.
The length of the new garage is 23.4 ft and the percentage increase in length is 81.6 %.
How to find original land new length of the square garage?Yossi used to have a square garage with 364ft of floor space.
Recently, built an addition to it.
The garage is still a square, but now it has 50% more floor space
Therefore,
area of a square = l²
where
l = side lengthTherefore,
364 = l²
l = √364
l = 19.0787840283
l = 19.1 ft
Therefore, the original length of the square garage is 19.1 ft.
After 50% increase in the area,
New area = 50 / 100(364) + 364
New area = 546 ft²
Therefore,
546 = l²
l = √546
l = 23.3666428911
l = 23.4 ft
Hence,
percentage increase in length = 19.1 / 23.4 × 100
percentage increase in length = 1910 / 23.4
percentage increase in length = 81.6239316239
percentage increase in length = 81. 6 %
Therefore, the length of the new garage is 23.4 ft and the percentage increase in length is 81.6 %.
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HELP ASAP PLEASE I need it this doesn’t make sense
Answers
Answer:
good luck i just answered for the 5 points
Step-by-step explanation:
fy(y)=0.2e^(-0.2y),y>=0
What is the probability that the buyer waits more than 10 minutes?
Answers
The probability that the buyer waits more than 10 minutes is 0.0271
How to determine the probability that the buyer waits more than 10 minutes?From the question, we have the following parameters that can be used in our computation:
Probability function:
fy(y)=0.2e^(-0.2y), y ≥ 0
The inequality expression y ≥ 0 means that
The value of y is atleast 0
The probability that the buyer waits more than 10 minutes means that
y = 10
So, we have the following representation
y = 10 in fy(y)=0.2e^(-0.2y)
Substitute the known values in the above equation, so, we have the following representation
fy(10)=0.2e^(-0.2 * 10)
Evaluate the exponents
fy(10)=0.2 * 0.13533528323
Evaluate the products
fy(10) = 0.0271
Hence, the probability is 0.0271
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Find the limit using l'Hôpital's Rule. Enter an exact numeric value. lim x→0 x tan(4x) =
(THE SOLUTION SHOULD BE DIVIDED INTO 2 PARTS, (1st PART )SUMMARY 2-3 LINES ONLY, (2nd PART) EXPLANATION CONTAINING 2-3 PARAGRAPHS.
Answers
Hey, are you sure the question should be x/tan(4x) since
the question you given does not need the LHoptial Rule to solve.
A = 11 + 2 (x-6) + 4 (-3-6)
Answers
Answer:
A = 2x - 37
Step-by-step explanation:
A
= 11 + 2(x - 6) + 4(-3 - 6)
= 11 + 2x - 12 - 12 - 24
= 2x - 37
What is the quotient of 2 divided by 2/3?
Answers
House of Mohammed sells packaged lunches, where their finance department has established a
weekly relationship between its Revenue R, in dollars and the number of lunches x, as R=x(82−x)
(a) Write the revenue R, in the form R=ax2+bx+c (b) What is the revenue when 35 lunches are sold? (c) Explain why the graph of R has a maximum.
(d) Write R in the form a(x−h)2+k
(e) How many lunches must be sold for them to achieve their maximum revenue?
(f) State the company's maximum revenue.
(2 marks) (2 marks) (2 marks)
(5 marks)
(2 marks) (2 marks)
(g) Complete the following table of values and graph the Revenue equation using the ordered pairs
(x , R). Label the coordinates of the vertex.
x
10
20
30
40
50
60
70
80
90
R
( 10 marks)
Answers
The revenue function is a quadratic equation and the graph of the function
has the shape of a parabola that is concave downwards.
The correct responses are;
(a) R = -x² + 82·x(b) $1,645(c) The graph of R has a maximum because the leading coefficient of the quadratic function for R is negative.(d) R = -1·(x - 41)² + 1,681(e) 41(f) $1,681Reasons:
The given function that gives the weekly revenue is; R = x·(82 - x)
Where;
R = The revenue in dollars
x = The number of lunches
(a) The revenue can be written in the form R = a·x² + b·x + c by expansion of the given function as follows;
R = x·(82 - x) = 82·x - x²
Which gives;
R = -x² + 82·xWhere, the constant term, c = 0
(b) When 35 launches are sold, we have;
x = 35
Which by plugging in the value of x = 35, gives;
R = 35 × (82 - 35) = 1,645
The revenue when 35 lunches are sold, R = $1,645
(c) The given function for R is R = x·(82 - x) = -x² + 82·x
Given that the leading coefficient is negative, the shape of graph of the
function R is concave downward, and therefore, the graph has only a
maximum point.
(d) The form a·(x - h)² + k is the vertex form of quadratic equation, where;
(h, k) = The vertex of the equation
a = The leading coefficient
The function, R = x·(82 - x), can be expressed in the form a·(x - h)² + k, as follows;
R = x·(82 - x) = -x² + 82·x
At the vertex, of the equation; f(x) = a·x² + b·x + c, we have;
\(\displaystyle x = \mathbf{-\frac{b}{2 \cdot a}}\)
Therefore, for the revenue function, the x-value of the vertex, is; \(\displaystyle x = -\frac{82}{2 \times (-1)} = \mathbf{41}\)
The revenue at the vertex is; \(R_{max}\) = 41×(82 - 41) = 1,681
Which gives;
(h, k) = (41, 1,681)
a = -1 (The coefficient of x² in -x² + 82·x)
The revenue equation in the form, a·(x - h)² + k is; R = -1·(x - 41)² + 1,681(e) The number of lunches that must be sold to achieve the maximum revenue is given by the x-value at the vertex, which is; x = 41
Therefore;
The number of lunches that must be sold for the maximum revenue to be achieved is 41 lunches(f) The maximum revenue is given by the revenue at the vertex point where x = 41, which is; R = $1,681
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you can speed from 0 to 25 m/s in 25 seconds and your friend can speed up from 0 to 6 m/s in 6 seconds. who is faster
Answers
Answer:
Neither; see below
Step-by-step explanation:
Your acceleration: \(a=\frac{\Delta v}{\Delta t}=\frac{25-0}{25}=1m/s^2\)
Your friend's acceleration: \(a=\frac{\Delta v}{\Delta t}=\frac{6-0}{6}=1m/s^2\)
Therefore, since both you and your friend accelerate at the same rate, neither is faster.
There are 25 trees on the Jackson’s property. Twenty percent of the trees are oak trees. Which equation can be used to find the number of oak trees on the property?
StartFraction 25 divided by 1 Over 20 divided by 1 EndFraction = StartFraction 25 Over 20 EndFraction
StartFraction 100 times 5 Over 20 times 5 EndFraction = StartFraction 500 Over 100 EndFraction
StartFraction 20 times 4 Over 25 times 4 EndFraction = StartFraction 80 Over 100 EndFraction
StartFraction 20 divided by 4 Over 100 divided by 4 EndFraction = StartFraction 5 Over 25 EndFraction
Answers
Answer:
StartFraction 25 divided by 1 Over 20 divided by 1 EndFraction = StartFraction 25 Over 20 EndFraction
Step-by-step explanation:
Number of trees = 25
Percentage of aok trees = 20%
To obtain the Number of oak trees :
(Number of trees ÷ 1) ÷ (percentage ÷1)
(25 / 1) ÷ (20 / 1) = (25 /1) * (20 / 1) = 25 / 20
StartFraction 25 divided by 1 Over 20 divided by 1 EndFraction = StartFraction 25 Over 20 EndFraction
A city is putting in a sand pit at the park. The length is an even, one-digit whole number, and
the width is 6.7 feet. Which explains the least and greatest possible areas for the sand pit?
Answers
Answer:
Step-by-step explanation:
Answer:
Listen the number is an EVEN one-digit, so that means all odd numbers that are one-digit numbers are out so that means number A is wrong so its between B, C, and D. If you read the answer throughully through B, C, and D you will find one of the 2 x 6.7 is wrong. To find the correct answer to 2 x 6 and if the sum of 2 x 6 is equal to your 2 x 6 you have the correct answer! Btw I forgot the exact answer but I know it was either B or C.
Step-by-step explanation:
An extrasolar planet is observed at a distance of
4.2 × 10⁹ kilometers away. A group of scientists
has designed a spaceship that can travel at the
speed of 7 × 108 kilometers per year. How many
years will the spaceship take to reach the extrasolar
planet? Enter the answer in the box.
Answers
After conducting some mathematical operations, we can conclude that it would take the spaceship 5555556 years to reach the extrasolar planet.
What do we mean by mathematical operations?A mathematical function known as an operation converts zero or more input values into a precisely defined output value.The quantity of operands affects the operation's arity.The four mathematical operations are functions that change one number into another by taking input values, or numbers, as inputs.They are addition, subtraction, multiplication, and division.So, years were taken by the ship to reach the extrasolar planet:
Distance of the planet: 4.2 × 10⁹ kmSpeed of the spaceship: 7 × 108 per/yearNow, calculate the number of years as follows:
= (4.2 × 10⁹)/(7 × 108)= (4.2 × 1000000000)/756= 4,20,00,00,000/756= 5555555.56Rounding off: 5555556 years
Therefore, after conducting some mathematical operations, we can conclude that it would take the spaceship 5555556 years to reach the extrasolar planet.
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The table shows three unique functions.
x f(x) g(x) h(x)
-2
4
6
-3
-1
1
2
1
4-
2
1
1
55 752
6
8
1
Hit
6:
Mark this and return
-25
Which statements can be used to compare the
characteristics of the functions? Select two options.
Of(x) has an all negative domain.
g(x) has the greatest maximum value.
All three functions share the same range.
Oh(x) has a range of all negative numbers.
All three functions share the same domain.
Answers
Answer:
The statements that can be used to compare the characteristics of the functions are:
1. g(x) has the greatest maximum value.
2. All three functions share the same domain.
Explanation:
- The table shows the values of three functions - f(x), g(x), and h(x) - evaluated at different values of x.
- We cannot determine the domain of f(x) or h(x) from the given table but we can see that g(x) has a domain of all real numbers.
- We can see that g(x) has the highest maximum value among the three functions, which is 8.
- We cannot determine the range of f(x) or g(x) from the given table but we can see that h(x) has a range of all negative numbers.
- We cannot say anything about the domain or range of f(x) based on the given table.
- Therefore, the two statements that can be used to compare the characteristics of the functions are: g(x) has the greatest maximum value and all three functions share the same domain.
Determine that the value x=9 of the following expiration given x2+3x-5
Answers
Step-by-step explanation:
To determine the value of x that satisfies the expression x^2 + 3x - 5 when x = 9, we simply substitute 9 for x and evaluate the expression:
9^2 + 3(9) - 5 = 81 + 27 - 5 = 103
Therefore, when x = 9, the expression x^2 + 3x - 5 evaluates to 103.
rate my answer, please.
Use the graph to answer the questions below.1284-6-4-2N468-12What is the domain of this function?
Answers
Answer:
The domain of the function are all real numbers between -1 and 0.
\(\lbrace x\in R\mleft|-1\leq x\leq0\rbrace\mright?\)Explanation:
The domain of function are set of values of x for which the function covers.
For the given function, from the graph the function starts from x =-1 and ends at x = 0.
So, the domain of the function are all real numbers between -1 and 0.
\(\lbrace x\in R\mleft|-1\leq x\leq0\rbrace\mright?\)I have provided a picture
Answers
The regression equation is y = 28.0 + 0.0510x.
The best predicted temperature at a time when the cricket chirps 3000 times in 1 minute, based on the regression equation is 181°F.
What is wrong with this predicted temperature: B. It is unrealistically high. The value 3000 is far outside of the range of observed values.
How to determine the line of best fit?In this scenario, the chirps in 1 min would be plotted on the x-axis (x-coordinate) of the scatter plot while the temperature would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.
From the scatter plot (see attachment) which models the relationship between the chirps in 1 min and the temperature, a regression equation or an equation for the line of best fit is given by:
y = 28.0 + 0.0510x
When x = 3000, the temperature can be calculated as follows;
y = 28.0 + 0.0510(3000)
y = 181°F
In conclusion, a temperature of 181°F is unrealistically too high because a chirps of 3000 times in 1 min is an outlier.
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An experiment was used to test a new migraine medicine. Each participant took either the new medicine or a placebo
then waited for one hour to see if the headache went away or remained. The results are compiled in the contingency
table below. Use the table to answer the questions.
Went Away| Medicine| Placebo
Remained|. 120. |. 68
18. | 44
Your answers should be exact numerical values.
There were
participants whose headache remained.
The probability of randomly selecting an individual whose headache remained is
There were
participants whose headache remained and took a placebo.
The probability of randomly selecting an individual whose headache remained and took a placebo is
Answers
1) Number of participants whose headache remained is: 62
2) The probability of randomly selecting an individual whose headache remained is: 0.4133
3) The probability of randomly selecting an individual whose headache remained and took a placebo is: 0.2933
How to find the probability of random selection?From the table, we have the parameters as:
Number of participants that took medicine and headache went away = 120
Number of participants that took placebo and headache went away = 68
Number of participants that took and headache remained = 18
Number of participants that took placebo and headache remained = 44
1) Number of participants whose headache remained = 18 + 44 = 62
2) The probability of randomly selecting an individual whose headache remained is:
62/(62 + 120 + 68)
= 62/150
= 0.4133
3) The probability of randomly selecting an individual whose headache remained and took a placebo is:
44/150 = 0.2933
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Expand & simplify
(x + 3) (x+ 1)
Answers
ZA and ZB are complementary angles. If mZA = (6x +29)° and m
ZB = (x + 5), then find the measure of ZA.
Answers
The following histogram shows the number of items sold at a grocery store at various prices: Histogram titled Items Sold with Price Range on the x axis and Number of Items Sold on the y axis. Bar 1 is 0 to 2 dollars and 50 cents and has a height of 2. Bar 2 is 2 dollars and 51 cents to 5 dollars and has a height of 4. Bar 3 is 5 dollars and 1 cent to 7 dollars and 50 cents and has a height of 0. Bar 4 is 7 dollars and 51 cents to 10 dollars and has a height of 1. Which of the following data sets is represented in the histogram? {0.50, 2.00, 2.52, 3.37, 4.53, 5.00, 8.99} {2, 4, 0, 1} {2.50, 2.51, 5.00, 5.01, 7.50, 9.00, 10.00} {0.50, 2.50, 2.50, 5.00, 5.00, 5.00, 7.51}
Answers
The second bar represents the price range from $2.51 to $5.00, so the data set includes the value of 4.53 and the value of 5.00 the correct answer is {0.50, 2.00, 2.52, 3.37, 4.53, 5.00, 8.99}
What is the histogram?A histogram is a graph used to represent the frequency distribution of a few data points of one variable. Histograms often classify data into various “bins” or “range groups” and count how many data points belong to each of those bins.
The histogram shows the number of items sold at various price ranges, and the heights of the bars indicate the number of items sold within each price range.
The data set that is represented in the histogram is:
{0.50, 2.00, 2.52, 3.37, 4.53, 5.00, 8.99}
This is because the histogram has four bars with heights of 2, 4, 0, and 1, which correspond to the number of items sold in each of the four price ranges shown.
The values in the data set correspond to the boundaries of these price ranges. For example, the first bar represents the price range from $0.00 to $2.50, so the data set includes the value 2.00 (which is within this range) and the value 2.52 (which is just outside this range).
Similarly, the second bar represents the price range from $2.51 to $5.00, so the data set includes the value of 4.53 (which is within this range) and the value of 5.00 (which is the upper boundary of this range).
Therefore, the second bar represents the price range from $2.51 to $5.00, so the data set includes the value of 4.53 and the value of 5.00 the correct answer is {0.50, 2.00, 2.52, 3.37, 4.53, 5.00, 8.99}
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Which of the following is an arithmetic sequence?
OA) 0, 5, 11, 18, 24, ...
B) -2, -4, -8, -16, -32, ...
OC) 0, 6, 12, 18, 24, ...
OD 1, -3, 9, -27, 81,...
Answers
Answer:
b. -2,-4,-8,-16,-32 thats the answer i think
The arithmetic sequences are:
B) -2, -4, -8, -16, -32, ...
C) 0, 6, 12, 18, 24, ...
The correct options are B and C.
What is an arithmetic sequence?A series of integers called an arithmetic progression or arithmetic sequence (AP) has a constant difference between the terms.
An arithmetic sequence is a sequence of numbers where each term is obtained by adding a fixed number, called the common difference, to the previous term. We can check which of the given sequences satisfy this definition:
A) 0, 5, 11, 18, 24, ...
The common difference between consecutive terms is not the same: the difference between the first two terms is 5, the difference between the second and third terms is 6, the difference between the third and fourth terms is 7, and so on. Therefore, this sequence is not an arithmetic sequence.
B) -2, -4, -8, -16, -32, ...
The common difference between consecutive terms is multiplied by -2: the difference between the first two terms is -2(-2) = 4, the difference between the second and third terms is -4(-2) = 8, the difference between the third and fourth terms is -8(-2) = 16, and so on. Therefore, this sequence is an arithmetic sequence with a common difference of -2.
C) 0, 6, 12, 18, 24, ...
The common difference between consecutive terms is 6. Therefore, this sequence is an arithmetic sequence with a common difference of 6.
D) 1, -3, 9, -27, 81,...
The common difference between consecutive terms is multiplied by -3: the difference between the first two terms is -3, the difference between the second and third terms is -3(-3) = 9, the difference between the third and fourth terms is -9(-3) = 27, and so on. Therefore, this sequence is not an arithmetic sequence.
Therefore, the arithmetic sequences are options B and C.
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