Use the summation formulas to rewrite the expression without the summation notation.
Answers
The expression without the summation notation is Sₙ = 4(n² - 1)/n²
How to rewrite an expression without the summation notation?
Summation notation is used to write a very long sum (of elements) in a very concise manner. It is also called Sigma notation
Given:
[tex]\displaystyle \sum\limits_{i = 1}^n \frac{12k(k-1)}{n^{3} }[/tex]
Below are the useful summation formulas you will need in order to rewrite the expression without the summation notation:
[tex]\displaystyle \sum\limits_{i = 1}^n i = \frac{{n\left( {n + 1} \right)}}{2}[/tex]
[tex]\displaystyle \sum\limits_{i = 1}^n {{i^2}} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}[/tex]
Where i is a variable
Since 12 and n are constants, we can rewrite the given equation as follows:
[tex]\displaystyle \sum\limits_{i = 1}^n \frac{12k(k-1)}{n^{3} } = \displaystyle \frac{12}{n^{3} } \displaystyle\sum\limits_{i = 1}^n k(k-1)[/tex]
[tex]= \displaystyle \frac{12}{n^{3} } \displaystyle\sum\limits_{i = 1}^n k(k-1)[/tex]
[tex]= \displaystyle \frac{12}{n^{3} } \displaystyle\sum\limits_{i = 1}^n (k^{2}-k)[/tex]
[tex]= \displaystyle \frac{12}{n^{3} } ( \displaystyle\sum\limits_{i = 1}^n k^{2} - \displaystyle\sum\limits_{i = 1}^n k )[/tex]
Using the formula, we have:
= 12/n³ ( n(n+1)(2n+1)/6 - n(n+1)/2 ) Factorise n(n+1) out
= 12n(n+1)/n³ ( (2n+1)/6 - 1/2 )
= 12(n+1)/n² ( (2n+1-3)/2 )
= 12(n+1)/n² ( (2n-2/6) )
= 2(n+1)/n² (2n-2)
= ( (2n+2)(2n-2) )/n²
= (4n² - 4)/n²
= 4(n² - 1)/n²
Therefore, using summation formulas to rewrite the expression without the summation notation gives Sₙ = 4(n² - 1)/n²
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Related Questions
I WILL GIVE BRAINLIEST!
Shamar belongs to a movie club that deducts $14 every month from his bank account. What is the change in Shamar's bank account after 7 months of paying for the movie club?
Answers
An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
The change in Shamar's bank account after 7 months is
x - 98.
Where x is the amount in the account initially.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
Amount in the account = x
Amount deducted every month = $14
The change in the account after 7 months.
= x - 7 x 14
= x - 98
Thus,
The change in Shamar's bank account after 7 months is
x - 98.
Where x is the amount in the account initially.
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A straw is placed inside a rectangular box that is 5 inches by 5 inches by 8 inches, as shown. If the straw fits exactly into the box diagonally from the bottom left corner to the top right back corner, how long is the straw? Leave your answer in simplest radical form. 8 in. 5 in. 5 in.
Answers
The length of the straw fitted in the box is 10.67 inches.
What is cuboid?A cuboid is a solid three-dimensional object which has six faces that are rectangular with eight vertices, and twelve edges.
Given that, a straw is placed inside a rectangular box that is 5 inches by 5 inches by 8 inches, the straw fits exactly into the box diagonally from the bottom left corner to the top right back corner,
Diagonal of a cuboid = √ l² + w² + h²
Therefore, diagonal of the box = √ 5² + 5² + 8² = 10.67 in
Hence, The length of the straw fitted in the box is 10.67 inches.
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David is three time as old as Joseph after 3 years David will be only twice as old as process will be there find their present ages
Answers
Answer:
Joseph's present age=3 yrs
David's present age=9 yrs
Step-by-step explanation:
Let Joseph's present age be x.
David's present age=3x
__________________
Ater 3 yrs,
Joseph's age=x+3
David's age=2(x+3)
Also, we can add 3 to David's present age to get age After 3 yrs. So, David's age after 3 yrs would also be 3x+3.
Now, balance them.
3x+3=2(x+3)
3x+3=2x+6
x=3
D"s prsent age=9
J's present age=3
Which question can be answered by finding the quotient of 5/6 divided by 1/20?
Answers
The question that can be answered by finding the quotient of 5/6 divided by 1/20 is C. Jared has 5/6 of an hour left to finish making bags. It takes him 1/20 of an hour's to make each bag. How many bags can he make?
How to illustrate the information?It should be noted that the quotient is the value that is gotten when we divide the numbers given.
In this case, Jared has 5/6 of an hour left to finish making bags. It takes him 1/20 of an hour's to make each bag.
The number of bags will be:
= 5/6 ÷ 1/20
= 5/6 × 20
= 100 / 6
= 17 bags
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Answer:
17 bags
Step-by-step explanation:
Which of the following is NOT true about the body mass index? A. uses statistical data of weight and height across the population B. measures the percentage of one's body fat directly C. relies on a correlation between weight and body fat D. does not account for one's body frame size or muscle content Please select the best answer from the choices provided. A B C D Mark this and return
Answers
The statement that is not true about body mass index is B. measures the percentage of one's body fat directly
What is Body mass index?The Body Mass Index (BMI) calculates a person's weight in relation to their height. It is not a precise assessment of a person's total body fat, more of an indicative.
The majority of the time, total body fat and BMI are correlated.
The Body mass index (BMI) formula
= weight in kilograms / square of height in meters
BMI Categories values are classified below
Underweight ≤ 18.5
Normal weight = 18.5–24.9
Overweight = 25–29.9
Obesity = BMI of 30 or greater
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Answer:B
Step-by-step explanation:
What is the correct answer
Answers
Answer:
given point is (-3,-2)
The rule for a rotation by 270° about the origin is (x,y)→(y,−x).
So,
(-3,-2)==============(-2,3).
there fore,
(-2,3) is the image point of (-3,-2)after the transformation r(x-axis) and rotation 270°.
Any confusion ask in CMT!!11 subtracted from the sum of 7y and 2y is -38.
Answers
Answer: 7y = 38.11111111111 2y = 10.8888888889 the sum of these two equals to 49 so 11-49=-38 Hopefully you understood. But I didn't know what you wanted to know from this?
Step-by-step explanation: 11-(7y+2y)=-38
7y+2y=9y (basic math)
11-9y=-38 then you want to move -38 to the other side, and when you move it to the other side it changes its value then from -38 it becomes 38. Same for the -9y, we move it to the other side and it changes its value so now from -9y its 9y. So now are equation should look like this 11+38=9y. 11+38=49 => y=49÷9 y=5.44444444444 (49÷9) x 7 = 38.1111111111 and (49÷9) x 2 = 10.8888888889
Exercise
1. If you have a case where n=25 and P=0.04, what is the probability that x is equal to 4?
Answers
If [tex]n=25[/tex] and [tex]p=0.04[/tex], then the probability of [tex]P(X=4)[/tex] is found to be 0.001374.
What is Binomial Distribution?
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a series of n independent experiments, each asking a yes-or-no question and having its own Boolean-valued outcome: success (with probability, [tex]p[/tex]) or failure (with probability, [tex]q=1-p[/tex]).
How to Find the Probability of a Binomial Distribution?
The chance of failure raised to the power of the difference between the number of tries and the number of successes is multiplied by the probability of success raised to the power of the number of successes in order to compute the binomial distribution. Next, multiply the final result by the sum of the number of attempts and the number of successes.The formula for binomial distribution is given as, [tex]P(X=k)={n \choose k}p^{k}(1-p)^{n-k}[/tex]From the given question, we have, [tex]n=25[/tex] and [tex]p=0.04[/tex]
Here, have to find the probability of [tex]P(X=4)[/tex], so [tex]k=4[/tex]
The binomial distribution formula is, [tex]P(X=k)={n \choose k}p^{k}(1-p)^{n-k}[/tex]
Substituting the given values in the above equation, we get
[tex]P(X=4)={25 \choose 4}0.04^{4}(1-0.04)^{25-4}[/tex]
[tex]\implies P(X=4)=12650 \times0.00000256\times0.4243\\\implies P(X=4)=0.001374[/tex]
Therefore, the value of [tex]P(X=4)=0.001374[/tex]
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List all possible rational roots for the given equation. x^4 - 2 x^2 + 16 x - 15 = 0
Answers
The possible roots for the given equation are, -1,3, -1-2i, -1+2i.
What do you mean by the roots of an equation?
The values of the variable that fulfill the equation are known as the roots of a quadratic equation. In addition, they are referred to as the "solutions" or "zeroes" of the quadratic equation. For instance, x = 2 and x = 5 satisfy the quadratic equation x2 - 7x + 10 = 0 and are hence the roots of the equation.
F(x) = x4-2x2-16x-15 has a root (zero) that you should find.
Finding x values for which F(x)=0 is the goal of the Polynomial Roots Calculator method set.
A tool from the list above is the rational roots test. Only numbers x that can be represented as the quotient of two integers, or rational roots, would be found.
According to the Rational Root Theorem, P and Q are factors of the Leading Coefficient and the Trailing Constant, respectively, if a polynomial zeroes for a rational number P/Q.
The Leading Coefficient is 1 and the Trailing Constant is -15 in this instance.
The component(s) are:
1 is the Leading Coefficient, while 1 is the Trailing Constant.
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Michel is taking 4 credit hours of courses at his local community college. Each credit hour costs $110.50. He is also to pay three technology fees of $19.25. Determine the total amount that he will pay in tuition.
Answers
Total: 499.75
Step-by-step explanation:
4 × 110.5 = 442
3 × 19.25 = 57.75
442 + 57.75 = 499.75
The equation for line t can be written as y+3= 1/2(x–7). Line u, which is perpendicular to line t, includes the point (3,3). What is the equation of line u?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.
Answers
The equation of a line u perpendicular to line t and passes through (3, -3) is y = -2x + 3
How to represent the slope of a line?The equation of a line can be represented in slope intercept form as follows:
y = mx + b
where
m = slopeb = y-interceptLine t has equation as y + 3 = 1 / 2 (x - 7).
Line u is perpendicular to line t and it passes through (3, -3).
The products of the slope of perpendicular lines is equals to negative 1.
Therefore,
m₁ m₂ = -1
1 / 2 m₂ = -1
m₂ = -2
Therefore, the equation of the line have a slope of -2.
let's find the y-intercept using (3, -3)
-3 = -2(3) + b
b = -3 + 6
b = 3
Therefore, the equation in slope intercept form is y = -2x + 3.
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What is the b-value?
Answers
Answer:
3
Step-by-step explanation:
y-intercept = b
The slope-intercept = y=mx+b
y=-2/5x+3
m=-2/5
b=3
If the equation was something like y=-2x-3, then the b would be -3 because you are subtracting 3.
2 step equation:12. Amy wants to start taking gymnastics. There is a one-time fee of $35 to sign up, then it costs $85 per month. If Amy saves up $225, how many months of gymnastics can she take?
Answers
The number of months Amy can take the Gymnastic = 2 months.
Amy wants to start taking gymnastics.
The fee structure of the gymnastics is divided into two parts.
There is a one time fee payment of 35$ and there is a monthly payment that costs 85$ per month.
So the first month fee of the gymnastics = 35 + 85 = 120$
Savings of Amy = 225$
If Amy pays the 1st month fee ,
then the she will be left with the savings = 225$ - 120$ = 105 $
It means Amy can join the gymnastics for one more month .
Hence the number of months Amy can take the Gymnastic = 2 months .
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If you roll a die 75 times. How many 1's should you expect to roll?
Answers
Do this: There are 6 sides on a die, one of the sides is a 1, since you roll it 75 times, do 75 / 6 which is 12.5
Which expression shows the distance you would drive from the library to the grocery
store along this road?
|-5+7)
-151+171
-5+7
1-5+171
Library Home
CLEAR
Grocery
Store
CHECK
Answers
Answer:
1-5+171
Step-by-step explanation:
Find the values of x and y
Answers
X= 20 & Y= 70 in right-angled triangle .
What is right angle triangle?
A right-angled triangle is a particular kind of triangle in which one of the angles is 90 degrees. The combined angles of the other two are 90 degrees. Perpendicular and the triangle's base make up the sides that the right angle is formed from. The longest of the three sides, known as the hypotenuse, is the third side.2X + 140 = 180
2X = 40°
X = 20°
X + Y = 90°
Y = 70°
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Apples cost 1.85 per kilogram
Work out the cost of 1.6 kilogram of apples
for 20 points
Answers
Answer:
1.6 kg = $2.96
Step-by-step explanation:
The given statement is,
→ 1 kg of apple = $1.85
→ a = 1.85
Now the cost of 1.6 kg of apple is,
→ 1.6 × a
→ 1.6 × 1.85
→ $2.96
Hence, 1.6 kg of apple is $2.96.
Identify the values of the variables. Give your answers in simplest radical form. Please help!
Answers
The values of the variables as represented in the right triangle are; (8√15)/3 and s = (4√15)/3.
What are the values of the variables in the given task content?It follows from the task content that the values of the variables in the right triangle are to be determined.
It follows from trigonometric ratios that;
Sin (∆) = opposite / hypothenuse
Cos (∆) = adjacent / hypothenuse
Tan (∆) = opposite / adjacent
It therefore follows from the attached image that;
Sin (60) = 4√5 ÷ p
p = 4√5 ÷ √3/2
p = (8√15)/3.
Also;
Tan (60) = 4√5 ÷ s
s = 4√5 ÷ √3
s = (4√15)/3
Therefore, the value of the variables are; p = (8√15)/3 and s = (4√15)/3.
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What other congruence is needed to prove that △QRP≅△QFP by ASA?
QR≅QF
∠R≅∠F
∠QPR≅∠QPF
PR≅PF
Answers
To prove that the two triangles are congruent by ASA, you need angles QPR and QPF to be congruent.
Answer = angle QPR congruent to angle QPF
X/-9 = 3
X equals what?
Answers
Answer:
X=-27
Step-by-step explanation:
X/-9 = 3
(-9) * X/-9 =(-9) * 3
X=(-9) * 3
X=-27
Answer:
x=-27
Step-by-step explanation:
x/-9=3
Multiply each side by -9
x=-27
That's it!
litterly help asap
The retail price of a golf club is $342. If the golf store has marked up the price by $42, what is the markup rate?
Answers
A golf club costs $342 at retail. The markup rate is 71.4% if the golf store has increased the price by $42.
What is meant by markup rate?The markup percentage is the difference between what an item costs the seller and what the customer pays. The greater the markup, expressed as a percentage of the cost, the greater the profit. While there is no "ideal" markup percentage, most businesses set it at 50%. A 50 percent markup, also known as a "keystone," means you are charging a price that is 50% higher than the cost of the good or service. Both terms are used to help determine profitability, but they are not the same! The markup is the percentage difference between a product's selling price and its cost.Given,
The difference in prices is $342 - $42 = $300
So, we need to find what percentage of 42 is 300
We do this by dividing: 300/42 = 7.14
which is 71.4 %.
So, the markup is 71.4 %.
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If V is the midpoint of WY, VY=29-4x, and WY=5x+19, find WV.
Answers
The value of WV is 9x - 10
What is line segment?A line segment can be defined as the part of a straight line which is bound by two distinct end points, and is known to be made up of every point on the line that between its endpoints.
From the information given, we can say that WY is a line with V as its midpoint with segments WV and VY.
We have;
VY=29-4x WY=5x+19This is represented as;
WV + VY = WY
Then;
WV = WY - VY
substitute the values
WV = 5x + 19 - (29 - 4x)
expand the bracket
WV = 5x + 19 - 29 + 4x
collect like terms
WV = 9x - 10
Hence, the value is 9x - 10
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I need help with solving problem.
Answers
Answer:
i can help but first what the questions and where are the options?
Step-by-step explanation:
Please help asap!! Please please please
Answers
Yes, I agree with Jorge.
Jorge's reasoning states that if two sides of the triangles are similar with a common measure of angle then one triangle can fit/overlap with the other triangle.
Jorge could improve his proof by stating his explanation in terms of mathematics.
He could have used the SAS similarity to show that the given triangles are similar.
What is the similarity test of two triangles?
If the ratio of the lengths of two sides of one triangle equals the ratio of the lengths of two sides of another triangle, and the included angles are equal, the two triangles are similar.
In the given figures, the length of side AB is equal to the length of side DE.
Side AB = Side DE ----------(I)
Similarly, the length of side BC is equal to the length of side EF.
Side BC = Side EF ----------(II)
And the measure of angle ABC is equal to the measure of angle DEF.
∠ ABC = ∠ DEF -------(III)
From (I), (II), and (III), and by using the SAS test we can say that
ΔABC ~ ΔDEF ----------- SAS test
Hence,
Yes, I agree with Jorge.
Jorge's reasoning states that if two sides of the triangles are similar with a common measure of angle then one triangle can fit/overlap with the other triangle.
Jorge could improve his proof by stating his explanation in terms of mathematics.
He could have used the SAS similarity to show that the given triangles are similar.
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Please solve I do not understand it
Answers
The expression [tex](\frac{1}{4^{-3 } } )^{-2}[/tex] is less than 1.
The sign to change to make the expression greater than 1 is changing the sign of the exponential value which will make the new expression as
[tex](\frac{1}{4^{-3 } } )^{2}[/tex]
What is expression?In maths, an expression is a combination of numbers, variables, functions (such as addition, subtraction, multiplication or division etc.).
In other words, an algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.)
Therefore, let's simplify the expression to know if it's equals to 1, greater than 1 or less than 1.
[tex](\frac{1}{4^{-3 } } )^{-2}[/tex]
using law of indices,
[tex](\frac{1}{4^{-3 } } )^{-2}[/tex] = 1 × [tex](4^{-3} )^{2}[/tex]
4⁻⁶ = 1 / 4⁶ = 1 / 4096
Hence,
1 / 4096 = 0.00024414062
Therefore, the value is less than 1.
Let's change one sign to make the expression greater than 1.
Therefore, the expression will be [tex](\frac{1}{4^{-3 } } )^{2}[/tex]
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The Finnish lottery draws 7 numbers from a total of 40 numbers.
What is the probability for a single lottery ticket to have 1 correct?
Answers
The probability that a single lottery has one correct number, using the hypergeometric distribution, is of:
0.4159 = 41.59%.
What is the hypergeometric distribution formula?The probability mass function of the hypergeometric distribution is presented as follows:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters of the function are listed as follows:
x is the number of successes in a single trial.N is the size of the population.n is the size of the sample selected in a single trial.k is the total number of desired outcomes.In the context of this problem, the values of these parameters are given as follows:
N = 40, k = 7, n = 7.
The probability that a single lottery has one correct number is P(X = 1), hence x = 1 and the calculation is as follows:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 1) = h(1,40,7,7) = \frac{C_{7,1}C_{33,6}}{C_{40,7}} = 0.4159[/tex]
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Rebecca wants to buy a new phone. She has already saved $110 and received a gift of $50 from her grandmother that she added to her savings. She plans to earn the rest by selling necklaces for $5 each. The phone she wants to purchase costs $485. Write an equation that represents how many necklaces Rebecca will need to sell in order to have enough money to get her phone.
Answers
Answer:
Step-by-step explanation:
She needs to sell 65 necklaces.
Given: ST=3x+3 and TU= 2x+9
What is the value of TU?
Answers
The value of TU is 21
From the question, we have
ST=3x+3 and TU=2x+9
ST = TU
3x + 3 = 2x + 9
3x - 2x = 9 - 3
x = 6
substituting the value, we get
TU = 2x + 9
= 2 (6) + 9
= 12 + 9
TU = 21
Addition:
The phrase "the addition" refers to combining two or more numbers. Adding two numbers is indicated by the plus sign (+), therefore adding three is written as three plus three. Additionally, the number of times the plus symbol (+) is used is up to you. For example, 3 + 3 + 3 + 3. Mathematical addition is a crucial component of the learning curriculum for kids. In primary school, students learn simple addition sums such as one-digit facts, Math addition sums, and double-digit Math addition sums. One of the most fundamental and interesting Math concepts for elementary school children is addition sums. Math is a fascinating subject, and children first learn basic mathematical concepts like adding numbers in their early years.
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0°F; drops 15° what’s the new temperature
Answers
The new temperature is given as -15°F.
What is a Measurement unit?A measurement unit is a unit to measure certain quantities.
For example, the mass can be measured in kilograms, length can be measured in meter and time can be measured in seconds.
To find the measurement of a new quantity known units can be used in operation. As to find the unit for speed we use m/s as a unit, where, m is the unit of distance and s is the unit of time.
Given that,
The present temperature is 0°F.
And, the drop in the temperature is 15°.
The new temperature after the drop can be found by subtracting the amount of drop from the initial temperature as follows,
0 - 15°F
= -15°F
Hence, the new temperature after the drop is -15°F.
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-353.92 - (-283.56) -131.29
Answers
-353.92-(-283.56)-131.29= -20,165