TRUE/FALSE. if january has 31 days, then 7 is an even number.

By using the induction, the equation 1+ 4 + 9 + ...........+n² = n(n-1)(2n-1)/6 where n ≥ 1 is proved.

The term induction in math is defined as a technique of proving a statement, theorem or formula which is thought to be true, for each and every natural number n.

In this case, we want to use induction to prove the equation:

=> 1 + 4 + 9 + ... + n² = n(n + 1)(2n + 1)/6 for all integers n ≥ 1.

To start the induction proof, we first prove that the statement is true for the base case n = 1.

The left side of the equation is 1, and the right side is

=> 1 * (1 + 1) * (2 * 1 + 1) / 6 = 1 * 2 * 3 / 6 = 1,

so the equation is true for n = 1.

Next, we assume that the equation is true for some integer k, where k ≥ 1. This means that:

=> 1 + 4 + 9 + ... + k² = k(k + 1)(2k + 1)/6.

Now, we use this information to prove that the statement is true for k + 1. The left side of the equation for k + 1 is:

=> 1 + 4 + 9 + ... + k² + (k + 1)².

This is equal to k(k + 1)(2k + 1)/6 + (k + 1)² by the induction hypothesis.

The right side of the equation for k + 1 is:

=> (k + 1)(k + 2)(2k + 3) / 6.

We can simplify this to:

=> (k² + 3k + 3) * (k + 2) / 6 = (k² + 3k + 3)(k + 2) / 6.

Finally, we need to show that these two expressions are equal:

=>k(k + 1)(2k + 1)/6 + (k + 1)² = (k² + 3k + 3)(k + 2) / 6.

Expanding both sides and cancelling common terms, we find that:

=> k³ + 3k² + 2k + k² + 3k + 3 = 2k³ + 9k² + 12k + 6.

This shows that the equation is true for k + 1, and by induction, it is true for all integers n ≥ 1.

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This is the vector equation equivalent to A = [2 7 -5 -1 2 -6 4 8], x = [2 -3], and b = [17 7 -22 16].

Simultaneous equations can be presented and solved more simply using matrix algebra, a type of mathematical notation. It can be used to generate a mathematical model of the structure and to provide a clear statement of a structural issue.

The vector equation is as follows

r = a(l – t)+ b t

where,

t is some parameter and

for points between A and B then 0≤t ≤ 1.

According to question:

The matrix equation [2 7 -5 -1 2 -6 4 8][2 -3] = [17 7 -22 16] can be written as a vector equation using the definition of the product of a matrix and a vector.

Let's call the matrix [2 7 -5 -1 2 -6 4 8] as matrix A and the vector [2 -3] as vector x. The product of matrix A and vector x is a new vector, which we'll call vector b.

The equation [2 7 -5 -1 2 -6 4 8][2 -3] = [17 7 -22 16] can be written as:

A * x = b

where,

A = [2 7 -5 -1 2 -6 4 8], x = [2 -3], and b = [17 7 -22 16].

This is the vector equation equivalent of the matrix equation, which represents the same linear relationship between the matrix, the vector, and the result vector.

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The side length of the square be 4/6 foot.

A square is a regular quadrilateral with four equal-length sides and four equal-length angles. The square's angles are 90 degrees or at right angles. In addition, the square's diagonals are equally spaced and split at a 90-degree angle.

Having four sides, a square is a quadrilateral. Each side measures the same length. A square has 360 degrees of angles total. In a square, every angle is a right angle.

Let the equation be

Area of a square = length²

If the area of a square is known, find the square root of the area to get the length.

√16 / 36 = 4/6 foot

Therefore, the side length of the square be 4/6 foot.

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Equivalent expression becomes -

(-3x² + x⁴ + x) + (2x⁴- 7 + 4x) = option: D

(x² - 2x) (2x + 3) = option: A

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It is possible to multiply, divide, add, or subtract in math. Any mathematical statement with variables, numbers, and an arithmetic operation between them is called an expression or an algebraic expression. For instance, 10m + 5 is an expression including the terms 10m and 5 as well as m, the variable of the supplied expression, all separated by the arithmetic sign "+".

Simplification:

= (-3x² + x⁴ + x) + (2x⁴- 7 + 4x)

= - 3x² + 3x⁴ + 5x - 7

= 3x⁴- 3x²  + 5x - 7

option : D

= (x² - 2x) (2x + 3)

= 2x³ + 3x² - 4x² - 6x

= 2x³ - x² - 6x

option: A

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To convert 200 meters to feet, you can use the following conversion factor 1 meter = 3.2808 feet

Meters and feet are units of length used to measure distances and lengths.

The conversion factor of 1 meter to 3.2808 feet is based on the definition of the foot as being equal to approximately 0.3048 meters.

To convert from meters to feet, simply multiply the number of meters by the conversion factor of 3.2808.

For example, 200 meters is equal to 200 x 3.2808 = 656.1664 feet.

It's important to note that this conversion factor is an approximation and may not be entirely accurate in all situations.

For more precise conversions, it's best to use a conversion calculator or a reference guide that provides exact conversion factors.

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Answer:

Below

Step-by-step explanation:

600 = cost

.08 * 600 = tax    

Add the two together    600 + .08 (600) = 1.08 * 600 = $ 648

The required they could have saved 21.97 feet of cable, as per the given conditions.

In a right-angled triangle, its sides, such as hypotenuse, and perpendicular, and the base is Pythagorean triplets.

Here,

Now, perpendicular p = 30 ft base b = 50 ft hypotenuse h = x ft

According to Pythagoras' theorem h² = p² + b²

x² = 50² + 30²

x = √2500 + 900
x = √3400
x = 58.3 feets

Wire saved = [30 + 50] - 58.3
Wire saved = 21.97

Thus, the required they could have saved 21.97 feet of cable, as per the given conditions.

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The milk will be served in cups to make it easier for them to consume, we are switching from the larger unit (i.e. quarts) to the smaller unit (i.e. cups).

4 cups or 2 pints are equivalent to one quart (qt). We can switch to utilizing gallons if we still need extra liquid. 16 cups, 8 pints, or 4 quarts make up a gallon (gal). It is a measurement of the largest liquid.

Servings per cup are divided at a rate of 9 servings to 3 cups, or 3 servings per cup. In a quart, there are 4 cups. In 2 quarts, there are 8 cups. In 4 quarts, there are 16 cups.

Servings per cup are divided at a rate of 9 servings to 3 cups, or 3 servings per cup.

1 quart = 4 cups

2 quarts = 2 × 4 quarts = 8 cups

3 servings/cup × 8 cups = 24 servings

The complete question is:

A recipe uses 3 cups of milk to make 9 servings. If the same amount of milk is used for each serving, how many servings can be made from two quarts?

1 gallon = 4 quarts

1 quart = 2 pints

1 pint = 2 cups

1 cup = 8 fluid ounces

Before you try that problem, answer the question below.

How many cups will you need to find the number of servings for?

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The expression if a = -2/7 than -a = 1/2 is a false statement.

The opposite of a number is obtained exchanging just the signal of the number.

The number for this problem is given as follows:

a = -2/7.

The signal is negative, hence the opposite of the number is given as follows:

-a = -(-2/7)

-a = 2/7. (two negatives become a positive).

As the problem states that the opposite is of -a = 1/2, the statement is false.

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Answer:

See the attached image.

Explanation:

Given the equation:

[tex]y=\left(\dfrac{1}{2}\right)^x + 1[/tex]

We can graph the top four points in the middle table, as they fit on the shown graph, and their coordinates are all integers:

[tex](-3,9)[/tex]   [tex](-2,5)[/tex]   [tex](-1,3)[/tex]   [tex](0,2)[/tex]

Then, we can draw a curve through the points and along the asymptote of the exponential function at [tex]y=1[/tex].

Note that the asymptote is drawn as a dotted line in purple.

The solution is after 3 yrs the populations of the two types of fish be the same.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

let, after t yrs the populations of the two types of fish be the same.

so, we get,

5750-250t = 3500+ 500t

2250=750t

t=3

Hence, The solution is after 3 yrs the populations of the two types of fish be the same.

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The solution of the expression is,

⇒ s = - 775

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

We have to given that;

The expression is,

⇒ s / - 31 = 25

Now, We can simplify as;

⇒ s / - 31 = 25

Multiply by - 31 both side,

⇒ s = - 31 × 25

⇒ s = - 775

Thus, The solution of the expression is,

⇒ s = - 775

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The answer is 12 or option (d)

I took the test~ good luck!

The volume of the solid is 16π/3 cubic units.

What is the volume of the solid?

The volume of a solid in mathematics is the measure of the space occupied by the solid. It can be calculated using methods such as the disk method or the shell method, depending on the shape of the region being rotated.

To find the volume of the solid obtained by rotating the region bounded by the curves y^2 = x and x = 2y about the x-axis, we would use the disk method. The disk method involves slicing the solid into infinitesimally thin disks, and adding up the volumes of these disks.

First, we need to find the equations for the curves y^2 = x and x = 2y. The first curve is already in the correct form, but the second curve can be rearranged to get y = x/2. Then, we can find the bounds of integration by finding the intersection of the two curves. These bounds are x = 0 and x = 4.

Next, we can set up the integral for the volume:

V = π ∫_0^4 (x/2)^2 dx

Evaluating this integral gives:

V = π/3 (x^3/2) evaluated from 0 to 4

V = π/3 [8/2 - 0/2]

V = π/3 (8) = 16π/3 cubic units

Hence, the volume of the solid is 16π/3 cubic units.

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The function that represents the context is y = 21x and  Zach's car can travel 336 miles using 16 gallons of gas.

A function can be defined as the outputs for a given set of inputs.

The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.

Given, Zach’s car travels 21 miles on 1 gallon of gas.

Let, x represents the number of gallons of gas and y represents the distance traveled.

Therefore, The function can be formed as,

y = 21x or f(x) = 21x.

The distance covered by Zach's car using 16 gallons of gas would be,

y = 21×16.

y = 336 miles.

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False. Span(u,v) includes linear combinations of both u and v.

The correct option is A

Now, According to the question:

When u and v are nonzero vectors, Span {u, v} contains only the line through u and the origin, and the line through v and the origin. b. Any list of five real numbers is a vector in ℝ5 .

Span(u,v) is the set of all linear combinations of u and v, meaning any combination of u and v multiplied by scalars. This includes scalar multiples of both u and v, but does not include the origin. To calculate Span(u,v), first write u and v as vectors u = (u1,u2,...,un) and v = (v1,v2,...,vn). Then, any linear combination of u and v can be written as c1u + c2v, where c1 and c2 are scalars, and the set of all such linear combinations is Span(u,v). For example, if u = (1,2) and v = (3,4), then Span(u,v) = {(1,2), (3,4), (4,6), (5,8), (7,10), ...}. In other words, Span(u,v) is the set of all scalar multiples of u and all scalar multiples of v.

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T symbol in statistics means Test statistic for t-test ( t-score )

What is  t-statistic ?

The t-statistic in statistics measures how far an estimated value of a parameter deviates from its hypothesised value in relation to its standard error.

T symbol in statistics mean Test statistic for t-test (  t-score )

Through the Student's t-test, it is utilised for testing hypotheses. To decide whether to accept or reject the null hypothesis in a t-test, the t-statistic is used.

It is quite comparable to the z-score, but when the sample size is small or the population standard deviation is unknown, the t-statistic is employed instead.

For instance, if the population standard deviation is unknown, the population mean can be estimated using the t-statistic from a sampling distribution of sample means.

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We need 32 rows in the truth table that would use to evaluate a logical statement in five variables.

It is TRUE.

Now, According to the question:

A truth table provides a method for mapping out the possible truth values in an expression and to determine their outcomes. The table includes a column for each variable in the expression and a row for each possible combination of truth values.

A truth table has one column for each input variable (for example, P and Q), and one final column showing all of the possible results of the logical operation that the table represents (for example, P XOR Q).

The formula to calculate the number of rows is equal to 2ⁿ, where n is the number of basic statement letters involved.

Number of rows = 2ⁿ

If the number of variables in a truth table is 5, then the number of rows = 2⁵

2⁵ = 2×2× 2× 2× 2

= 32

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Answer:

P = 24.997 cm

Step-by-step explanation:

the perimeter (P) is the sum of the 2 radii and the arc

the arc length is calculated as

arc = circumference of circle × fraction of circle

      = 2πr × [tex]\frac{90}{360}[/tex] ( r is the radius , here r = 7 )

      = 2 × 3.142 × 7 × [tex]\frac{1}{4}[/tex]

      = 43.988 ÷ 4

      = 10.997 cm

Then

P = 10.997 + 7 + 7 = 24.997 cm

A cubic close-packed (ccp) unit cell has a coordination number of 12.

A cubic close-packed (ccp) unit cell has an arrangement of atoms where each atom is surrounded by 12 other atoms, resulting in a coordination number of 12.

A cubic close-packed (ccp) unit cell is a type of unit cell that has an arrangement of atoms where each atom is surrounded by 12 other atoms. This arrangement forms a structure with a coordination number of 12, meaning that each atom has 12 other atoms in contact with it. This type of arrangement is very common in many materials, and is often used to describe the structure of metals and other crystalline materials. The atoms in a ccp unit cell are arranged in a regular pattern, with three atoms in each of the eight corners and four atoms in the center of each face of the unit cell.

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The shortest distance between Tim and Jan is given as follows:

19.2 miles.

(they are 19.2 miles apart).

Jonathan's mistake is that the shortest distance is not obtained passing through the home, but walking diagonally from Tim's position to Jonathan's position or vice versa.

The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.

To obtain the shortest distance between Tim and Jan, we consider a right triangle in which:

Hence the distance is obtained as follows:

d² = 12² + 15²

d = square root(12² + 15²)

d = 19.2 miles.

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The proof that can be used to show that this is an isosceles triangle has been given below

A cyclic quadrilateral is a quadrilateral whose vertices lie on the same circle. In this case, since ABCD is a cyclic quadrilateral and arc AD = arc DC, it follows that the opposite angles of the quadrilateral are equal. Let's call the equal angles of the quadrilateral "x."

Since AB = CE, we can use the angle-angle criterion for congruent triangles to prove that triangle ABD is congruent to triangle ECD:

ADB and CDE are both right angles, and the equal angles x in triangle ABD and CDE both add up to 90 degrees.

So, by the angle-angle criterion, triangle ABD is congruent to triangle CDE.

Since AB = CE and ABD is congruent to CDE, we can conclude that triangle ABD is isosceles.

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Answer:

no solution is the answer

Using the formula of compound interest and simple interest, the investment option that is better is at interest rate of 3.99% which is compounded monthly

To compare the investment options, you need to calculate the future value of each investment after 8 years. Here's a breakdown of each option:

A. 3.99% compounded monthly: This option earns interest every month and compounds, meaning the interest is added to the principal, so future interest is earned on the increased principal amount. This type of interest compound is the most advantageous to the investor. The formula for calculating the future value for this option is:

FV = P * (1 + r/n)^(nt)

FV = 12000(1 + 0.0399 / 12) * (12 * 8)

FV = $16,503.58

where P is the principal ($12,000), r is the annual interest rate (3.99%), n is the number of times the interest is compounded in a year (12 times), t is the number of years invested (8 years).

B. 4% compounded quarterly: This option earns interest every quarter and compounds. The formula for calculating the future value is the same as option A, with the only difference being the number of times the interest is compounded in a year (n=4).

Substituting the values into the formula;

FV = $16,499.29

C. 4.175% compounded annually: This option earns interest once a year and compounds. The formula for calculating the future value is the same as options A and B, with the only difference being the number of times the interest is compounded in a year (n=1).

Substituting the values into the formula

FV = $16,645.21

D. 4.2% simple interest: This option earns interest once a year based on the original principal amount and does not compound. The formula for calculating the future value for this option is:

FV = P(1 + rt)

FV = 12000 (1 +  0.042 * 8)

FV = $16032

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Answer:

The Amplitude is 3

Step-by-step explanation:

How that helps

Answer:

The amplitude of this function is 4, which is calculated by subtracting the minimum value (1) from the maximum value (5).

Step-by-step explanation:

Answer:

105 pages

Step-by-step explanation:

We can solve this using ratios

Changing 3 hours to minutes

3 * 60 = 180 minutes

7 pages            x pages

---------------   = ----------------

12 minutes         180 minutes

Using cross products

180 * 7 = 12 x

Divide each side by 12

180*7/12 = x

105 =x

105 pages

Answer:If you can read 7 pages in 12 minutes then you will be able to read 105 pages in 3 hours.

Step-by-step explanation: you multiply 12x15 to get 180 since 180 is 3 hours in minutes then you multiply 7x15= 105. So you will read 105 pages every 3 hours.

100th Answer!!!!

A. The price of the telephone after a 30% discount will be $27.986.

B. The discount in the form of a percentage will be 41.02%.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.

The percentage is given as,

Percentage (P) = [Final value - Initial value] / Initial value x 100

Woolworths had a going-out-of-business sale. The price of a telephone before the sale was $39.98.

A.  The  price of the telephone after a 30% discount will be given as,

⇒ $39.98 x (1 - 0.30)

⇒ $39.98 x 0.70

⇒ $27.986

B.  If the sale price of the same telephone had been $23.99. Then the percentage is given as,

P = [(39.98 - 23.99) / 39.98] x 100

P = (15.99 / 38.98) x 100

P = 0.4102 x 100

P = 41.02%

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The end behavior of the function f(x) is defined as follows:

As x -> - ∞, f(x) -> ∞. As x -> ∞, f(x) -> 0.

The function for this problem is defined as follows:

y = 1.5(0.8)^x.

(converting the fractions to decimals).

The end behavior of a function is given by the limit of the function is the input x goes to either negative infinity or positive infinity.

When x -> -∞, (0.8)^(-∞) = (5/4)^∞ -> ∞, hence:

As x -> - ∞, f(x) -> ∞.

When x -> -∞, (0.8)^(∞) -> 0, as it has a base value with absolute value less than 1, hence:

As x -> ∞, f(x) -> 0.

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The height of the next bounce which is on the third bounce will be 12.8 feet.

A series of non-zero integers where every term after the first is obtained by increasing the one before it by a constant, non-zero value known as the scale factor.

Let a₁ be the first term and r be the common ratio.

Then the nth term of the geometric sequence is given as,

aₙ = a₁ · (r)ⁿ⁻¹

Then the common ratio is given as,

r = 80 / 200

r = 0.40

The height of the next bounce will be given as,

a₄ = 200 · (0.4)⁴⁻¹

a₄ = 200 · (0.4)³

a₄ = 200 · 0.064

a₄ = 12.8 feet

The height of the next bounce which is on the third bounce will be 12.8 feet.

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Tom's increased monthly rent after the 8.5% increase will be R1356.25.

From the question, we have the following parameters that can be used in our computation:

Monthly rent = R1250

Yearly rate = 8.5%

If Tom pays R1250 every month, then his annual rent is:

Annual rent = 1250 * 12 = R15000.

If his annual rent increased by 8.5%, his new annual rent will be

New = 15000 * (1 + 0.085) = 16275.

Divide by 12 to get the increased monthly rent

Increased monthly rent = 16275 / 12 = R1356.25.

Hence, the rent is R1356.25.

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