The formula V = s3 give the volume V of a cube with side length s.

Step-by-step explanation:

We're asked to find the simplified form of,

[tex]\longrightarrow S=\underbrace{1^2+3^2+5^2+7^2+\,\dots}_{\textrm{$n$ terms}}[/tex]

The n'th term of the sequence [tex]1^2,\ 3^2,\ 5^2,\ 7^2,\,\dots[/tex] is [tex](2n-1)^2.[/tex] So,

[tex]\longrightarrow S=1^2+3^2+5^2+\,\dots\,+(2n-1)^2[/tex]

We know the following sum,

[tex]\longrightarrow 1^2+2^2+3^2+\,\dots\,+n^2=\dfrac{n(n+1)(2n+1)}{6}[/tex]

This is expression for the sum of squares of first 'n' natural numbers. In case of '2n' natural numbers,

[tex]\longrightarrow 1^2+2^2+3^2+\,\dots\,+(2n)^2=\dfrac{2n(2n+1)(4n+1)}{6}[/tex]

We separate odd squares and even squares in the LHS.

[tex]\begin{aligned}\longrightarrow\ \ &\Big(1^2+3^2+5^2+\,\dots\,+(2n-1)^2\Big)&\\+\ \ &\Big(2^2+4^2+6^2+\,\dots\,+(2n)^2\Big)&=\dfrac{2n(2n+1)(4n+1)}{6}\end{aligned}[/tex]

We can take 4 common from the sum of even squares.

[tex]\begin{aligned}\longrightarrow\ \ &\Big(1^2+3^2+5^2+\,\dots\,+(2n-1)^2\Big)&\\+\ \ &4\Big(1^2+2^2+3^2+\,\dots\,+n^2\Big)&=\dfrac{2n(2n+1)(4n+1)}{6}\end{aligned}[/tex]

Now the sums in LHS are, as written above,

[tex]1^2+3^2+5^2+\,\dots\,+(2n-1)^2=S[/tex]

[tex]1^2+2^2+3^2+\,\dots\,+n^2=\dfrac{n(n+1)(2n+1)}{6}[/tex]

So,

[tex]\longrightarrow S+\dfrac{4n(n+1)(2n+1)}{6}=\dfrac{2n(2n+1)(4n+1)}{6}[/tex]

[tex]\longrightarrow S=\dfrac{2n(2n+1)(4n+1)}{6}-\dfrac{4n(n+1)(2n+1)}{6}[/tex]

[tex]\longrightarrow S=\dfrac{2n+1}{6}\Big[2n(4n+1)-4n(n+1)\Big][/tex]

[tex]\longrightarrow S=\dfrac{2n+1}{6}\Big[4n^2-2n\Big][/tex]

[tex]\longrightarrow S=\dfrac{2n(2n-1)(2n+1)}{6}[/tex]

[tex]\longrightarrow\underline{\underline{S=\dfrac{n}{3}\,(4n^2-1)}}[/tex]

This is the expression for the sum of squares of first 'n' odd natural numbers.

We will use the concept of Probability in this question. The probability that Hannah will win both games is 0.42.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty. The likelihood that an event will occur increases with its probability. A straightforward illustration is tossing a fair (impartial) coin. Since there are no other conceivable outcomes and the coin is fair, the odds of both the possibilities, "heads" and "tails," are equally likely to occur. As a result, the probability of either occurrence is half.

The probability she will win the game of chess is 0.6.

The probability she will not win  the game of chess is 0.4[1-0.6]

The probability that she will win the game of backgammon is 0.7.

The probability that she will not win the game of backgammon is 0.3.[1-0.7]

To find the probability of winning of two games

    P(A∩B)=P(A)×P(B)

   ⇒P(Hannah)=0.6×0.7

    ∴Probability of winning both games=0.42.

We will use the concept of Probability in this question. The probability that Hannah will win both games is 0.42.

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The most important step in financial planning is to create a budget.

What is financial planning?

The process of setting and achieving long-term financial goals is called financial planning. It involves evaluating one's current financial resources, setting financial objectives, and devising a plan to achieve those objectives. It also entails developing savings, managing debt, and creating a budget.

According to the given question:

Financial planning is crucial because it enables individuals and families to better manage their finances and ensure that they will be able to achieve their desired objectives. Additionally, it aids individuals in making sound financial decisions and preventing costly errors.

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The median is the middle score in the set.

To find the median, start by arranging all of the data points from smallest to largest.

In an odd-numbered set, the median will be the number in the very middle of the list.

In an even-numbered set, you will need to calculate the average of the two middle numbers.

To do this:

Step 1: Take the two middle numbers of the even-numbered set

Step 2: Add the two numbers together

Step 3: Divide the total by 2

As an example, consider this set of numbers: 5, 9, 11, 9, 7. First, you arrange them in numerical order (5, 7, 9, 9, 11). Since you have an odd number of scores, the number in the third position of the data set is the median which, in this case, is 9 (5, 7, 9, 9, 11).

To calculate the median for an even number of scores, imagine that your research revealed this set of data: 2, 5, 1, 4, 2, 7. Your first step is to put them in numerical order (1, 2, 2, 4, 5, 7). The two middle scores are 2 and 4, so you should add them together (2+4=6) and then divide 6 by 2, which equals 3. In this data set, the median score is 3.

The median is calculated by arranging the scores in numerical order, dividing the total number of scores by two, then rounding that number up if using an odd number of scores to get the position of the median or, if using an even number of scores, by averaging the number in that position and the next position.

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The SAS Similarity Theorem asserts that two triangles are similar if two sides in one triangle are proportionate to two sides in another triangle and the included angle in both is congruent.

The side-angle-side (SAS) theorem is one such example. Triangles are congruent if their two sides and included angles are equivalent to those of another triangle's two sides and included angle. SAS similarity analysis: When the ratio of two sides' lengths in one triangle equals the ratio of sides' lengths in another triangle, and the included angles are also equal, the two triangles are said to be comparable.

Area of triangle =

[tex]1/2 X b X h\\b= 10\\h=5\\A = 1/2 X 10X 5\\A = 25 unit sq.[/tex]

and also by heron's formula

Area = [tex]\sqrt{s(s - a)(s - b)(s - c) }[/tex]

and, here s = half the perimeter,

[tex](a + b + c)/2.[/tex]

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Write a function's inverse as y = f(x), then solve for x as a function of y to discover the inverse of the function.

Multiplying all of the terms on the equation's left and right sides by the fraction's denominator will remove a fraction from it. Try to solve for the variable. Isolate the fractional word initially. By multiplying both sides of the equation by the reciprocal of 4 / 7 (4/7), you can now eliminate the fractional coefficient.

A rational function in mathematics is any function that can be expressed as a rational fraction, which is an algebraic fraction in which the denominator and numerator are both polynomials. In particular, a polynomial term cannot contain any square roots of variables, fractional or negative powers on the variables, or variables in any fraction's denominator.

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The price of a computer increase from £631 to £776. 13. The percentage change is 81.3%

As per the given data:

Old value of computer is £631

New value of computer is £776. 13

Percentage Change: Percentage Change is the difference coming after subtracting the old value from the new value and then divide by the old value and the final answer will be multiplied by 100.

The phrase percent change says the difference in the percent of the earlier and present quantity.

Percentage change formula is

Percentage Change [tex]=\frac{{New Value - Old Value} }{{Old Value}} \times 100[/tex]

If the result is positive, then it is an increase.

If the result is negative, then it is a decrease.

[tex]=\frac{631}{776.13} \times 100[/tex]

= 0.813 × 100

= 81.3%

Therefore the increase in percentage is 81.3%

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Therefore ,  we take the square root of both sides as if  two expressions are equal, then their square roots are also equal.

The number is derived by multiplying the square root of a positive integer by itself.

For instance, the square root of 25 is 5, since 5 x 5 is 25.

As you can see, 5 is also the square root of 25 because (5) (5) = 25.

The square root of a number is always represented by the symbol, which is √.

Here,

We are using the square to solve a quadratic problem

Remember to take the square root of both aspects of the equation and to utilize both negative and positive roots.

Because of the rule that says if expressions are equivalent, then their square roots are likewise equal, we are able to perform this.

Therefore ,  we take the square root of both sides as if  two expressions are equal, then their square roots are also equal.

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Answer:

For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.

True. If y is a differentiable function of u and u is a differentiable function of x, then the chain rule states that y is a differentiable function of x.

The chain rule states that if y is a function of u and u is a function of x, then the derivative of y with respect to x is equal to the derivative of y with respect to u multiplied by the derivative of u with respect to x. This means that if y is a differentiable function of u and u is a differentiable function of x, then y is also a differentiable function of x.

In mathematics, a function is said to be differentiable at a point if it is possible to define the derivative of the function at that point. The derivative of a function is a measure of how the function changes as its input (or independent variable) changes. It is defined as the limit of the difference quotient, which is a measure of the rate of change of the function over a small interval around the point of interest.

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If Last year, Milan opened an investment account with $5200. At the end of the year, the amount in the account had decreased by 7%.  the year-end amount in terms of the original amount is $364.00 and the year-end amount in Milan's account is $4,836.00.

a. Year end amount in terms of original amount

Year end amount = Principal × Percentage decrease

Year end amount = $5,200 × 7%

Year end amount = $364.00

b. Year end amount in Milan account

Year end amount = $5,200 -$364

Year end amount = $4,836.00

Therefore the year-end amount in terms of the original amount is $364.00 and the year-end amount in Milan's account is $4,836.00.

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There are four factors that should be taken into consideration when creating a financial plan.

The process of figuring out how a corporation will be able to afford to meet its strategic goals and objectives is known as financial planning. Typically, a business develops a financial plan right away after establishing its vision and goals.

You can reach your financial objectives with the help of financial planning, whether they are to purchase a home for your family, save money for your children's school, have a comfortable retirement, or take a dream vacation. Additionally, it gets you ready for crises and unanticipated circumstances like getting sick, losing your job, or needing home renovations.

4 things to think about when creating your financial plan:

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Answer:

Step-by-step explanation:

Answer:

don't know the answer

Step-by-step explanation:

but thanks for the points bookie

Standard deviation of 13 denotes a high value. This indicates observed data values are dispersed in relation to mean value.

Standard deviation expresses the amount of scatter of a set of values.

It is an estimation that implies how far each data value apart from the mean value. A high deviation scores denotes that the data are dispersed around the mean value. A low value indicates that all data value is close in relation to the mean value.

As per the statistical rules, any standard deviation value above or equal to 2 is considered as high. So, this value is high. According to the fourth-grade test for insurance policy, score 13 is equivalent to the 84th percentile. And as per the fifth-grade test score 13 is equivalent to the 98 percentiles.

In accordance with the empirical assumption of normal distribution, this higher value of standard deviation implies most of the data will be dispersed in relation to the mean value.

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Answer:

This data has a positive correlation.

Step-by-step explanation:

As the x-values increase, the y-values increase. Therefore, there is a positive correlation. (positive/positive = positive)

The total surface area of the rectangular prism in the figure is 348 cm²

A rectangular prism is a polyhedron in geometry that has two parallel and congruent bases.

It also goes by the name cuboid. Six faces make up a rectangular prism, and each face has a rectangle shape and twelve edges.

The formula for total surface area of a rectangular prism is

TSA = 2 (lh +wh + lw ) Square units

= 2( 8 *9 + 6 * 9 + 8 * 6 )

= 2 (174)

= 348 cm²

The rectangular prism have a Total surface area of  348 cm²

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Answer:

x=13

Step-by-step explanation:

the bottom line is 11 the top line is 6 the difference is 5 hence we use the the formula 12 5 13 i mean the pythogrious formula

Answer:

12x plus 9x

Step-by-step explanation:

Answer:

the second

Step-by-step explanation:

so first you now that slope is y=mx+b with b=y, so with this we go (0,-3) down the the slope is 2 so we go up two right one the the opposite down two left then do that a couple of times and you have your line. since Y is less than the coloring goes down. HOPE THIS HELPS

We can determine 3 terms in the expression 3×2y, 3 terms in the expression 2y2z and 3 terms in the expression z2×5

An expression is the symbolic form of representing a mathematical statement. In a word, description of a mathematical operation is expressed by a few terms and symbols. Expression has two component terms and symbols in where terms mean number, number accompanied by variables and variables. On the other hand, symbols mean addition, subtraction, division, multiplication, equal, inequal sign and so on.

In the question, we notice there are three expressions. The first one, 3 × 2y

3 is multiplied by 2y, here 3 is numbers and it is called constant, y is a variable and 2 is also a number but coefficient of y.

In the second expression, 2y2z, there are 3 terms, but number 2 is used in two times. In both cases, 2 is a coefficient of y and z respectively.

In the last, z2 × 5, we find there are 3 terms. 2 is a number called coefficient of variable z and 5 is constant.

we can say 2z is multiplied by 5.

hence, we find total 6 terms x, y, z, 2, 3 and 5.

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The scale factor from the cliff in the Scale Drawing to the actual cliff is 3, and the actual height of the lighthouse is 30 ft.

To adjust the size of a figure without altering its shape, a scale factor is a number or conversion factor that is utilized. It is employed to change the size of an object.

Given:

The actual height from sea to the lighthouse, A = 36ft

The height in the diagram, D = 12

Calculate the scale factor as shown below,

Scale factor = A / D

SF = 36 / 12

SF = 3

Calculate the actual height as shown below,

Actual height = scale factor × height in the diagram

Actual height = 3 ×  10 = 30 ft.

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We can conclude that under the AAS congruency condition, we can prove that △WXY ≅ △YZX.

Two triangles are said to be congruent if all three corresponding sides and all three corresponding angles have the same size.

You can move, flip, twist, and turn these triangles to produce the same effect.

When relocated, they are parallel to one another.

So, we know that;
Shared side XZ by △WXZ and △YZX.

Right angles exist between WXZ and XZY.

∠XYZ and ∠XYZ have the same values.

Now that we have the parameters, we can infer that XZ is congruent to itself according to the reflexive property of congruence, and as a result, both triangles have a congruent side.

Second, since WXZ and XZY are right angles, it follows that they are equal and hence congruent.

The last piece of information is that XWZ and XYZ are congruent.

In conclusion, the triangles are congruent according to the AAS Congruency since we have two corresponding angles and one corresponding side that are equal but the corresponding side is not with the included angles.

Therefore, we can conclude that under the AAS congruency condition, we can prove that △WXY ≅ △YZX.

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Step-by-step explanation:

13 cups = ? pints

there are 2 cups per pint so:

13/2 = 6.5

13 cups = 6.5 or [tex]6\frac{1}{2}[/tex] pints

Kim could start contributing to her employer-sponsored 401 K for retirement and 529 plans for her son's college education.

401(k) is an employer sponsored retirement benefits plan where the employer and employee contribute a percent of the employee's benefits or gross income every payday in order that the funds can be used to to generate further wealth by a way of investment for the benefit of the employees upon retirement.

529 plan on the other hand,is a college savings plan wherein parent,guardian or even the beneficiary(the would college student) saves money at interval with hope of funding the costs of college education

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Answer: 3.5

Step-by-step explanation:

To solve for x in a square, you need to find the value of x that makes the square equal to zero. In this case, the square is given by (6x-21), so we can solve for x by setting this expression equal to zero and then solving for x.

To solve the equation (6x-21) = 0, we can use basic algebraic techniques. First, we can add 21 to both sides of the equation to get (6x-21) + 21 = 21. Then, we can use the distributive property to expand the left-hand side of the equation as follows:

(6x-21) + 21 = 21

6x - 21 + 21 = 21

6x = 21

Finally, we can divide both sides of the equation by 6 to find the value of x that makes the square equal to zero:

6x = 21

x = 21 / 6

x = 3.5

Therefore, the value of x that makes the square (6x-21) equal to zero is x = 3.5.

The expression represents the probability that both students chosen are sophomores is C( 6 , 1 ) * C ( 5 , 1 ) / C ( 20 , 2 ) .

Given :

The requested probability can be computed as the ratio between the number of ways to choose two sophomores in alternate positions [tex]N_s[/tex] and the total number of possible choices [tex]N_t[/tex] .

p = [tex]N_s[/tex] / [tex]N_t[/tex]

There are 6 sophomores and 14 freshmen to choose from each separate set. There are 20 students in total .

We'll assume the positions of the selections are NOT significative, i.e. student A/student B is the same as student B/student A.

To choose 2 sophomores out of the 6 available, the first position has 6 elements to choose from, the second has now only 5.

C( 6 , 1 ) * C ( 5 , 1 ) ways .

The total number of possible choices is C ( 20 , 2 )

Probability = C( 6 , 1 ) * C ( 5 , 1 ) / C ( 20 , 2 )

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Full question :

Six sophomores and 14 freshmen are competing for two alternate positions on the debate team. Which expression represents the probability that both students chosen are sophomores? StartFraction (6 C 1) (5 C 1) Over 20 C 2 EndFraction StartFraction (6 P 1) (5 P 1) Over 20 P 2 EndFraction StartFraction (20 C 6) (19 C 5) Over 20 C 2 EndFraction StartFraction (20 P 6) (19 P 5) Over 20 P 2 EndFraction

In math for kids, a set is a collection of objects or numbers that have been grouped together.

Set is a collection of distinct elements or objects that are grouped together. Sets can be defined as either finite (having a finite number of elements) or infinite (having an infinite number of elements). Sets are commonly used in mathematics, computer science and other areas of study to represent collections of data or to express logical relationships.

The objects in a set can be anything, such as numbers, shapes, letters, words, and even other sets. Sets are often used in math problems and are a great way to help kids learn basic math skills. They can also be used to teach problem solving and logic. Sets are a fundamental part of mathematics and can be used to help kids understand mathematical concepts.

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