omework: 2.1 tion list Solve the following equation and check. (x)/(7)+(x)/(9)=(9)/(14)

To prove that \[ A^{n}=\left[\begin{array}{ccc} a^{n} & n a^{n-1} & \frac{n(n-1)}{2}a^{n-2} \end{array}\right], we will use mathematical induction.

Base case: For \( n=1 \), \( A^{1}=A \). This is true since
\[ A=\left[\begin{array}{ccc} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{array}\right] \]
which matches the left side of our equation.

Inductive step: Assume that the statement is true for some \( n=k \), i.e.
\[ A^{k}=\left[\begin{array}{ccc} a^{k} & ka^{k-1} & \frac{k(k-1)}{2}a^{k-2} \end{array}\right] \]
We will now show that it is true for \( n=k+1 \).
\[ A^{k+1}=A^{k}\cdot A \]

We substitute the induction hypothesis into this equation and get:
\[ A^{k+1}=\left[\begin{array}{ccc} a^{k} & ka^{k-1} & \frac{k(k-1)}{2}a^{k-2} \end{array}\right]\cdot \left[\begin{array}{ccc} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{array}\right] \]

Calculating the matrix product yields:
\[ A^{k+1}=\left[\begin{array}{ccc} a^{k+1} & (k+1)a^{k} & \frac{(k+1)k}{2}a^{k-1} \end{array}\right] \]
which matches the left side of our equation, completing the proof.

Learn more about  completing the proof

brainly.com/question/30920773

#SPJ11

5/2 is the slope of line .

What is slope, exactly?

Typically, a line's slope provides information about the steepness and direction of the line. Finding the difference between the coordinates of the locations will allow you to quickly compute the slope of a straight line connecting two points, (x1,y1) and (x2,y2).

                           The letter "m" is frequently used to signify slope. A line's steepness can be determined by its slope. In mathematics, slope is calculated as "rise over run" (change in y divided by change in x).

points (2,6) and (0,1)

  Slope = 1 - 6/0 - 2

             = -5/-2

             = 5/2

Learn more about slope

brainly.com/question/3605446

#SPJ9

Answer:

The answer is  3

The answer would be 51,998

Answer:

The formula for calculating the amount of money in an account with continuous compounding is:

A = Pe^(rt)

Where:

A = the amount of money in the account after t years

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

t = the time (in years)

Substituting the given values into the formula, we get:

A = 7500e^(0.0158)

A ≈ $9,071.21

Therefore, to the nearest hundred dollars, there would be $9,100 in the account after 8 years.

Step-by-step explanation:

Martin buys a total of 2(5)/(6) + 4(3)/(4) = 10/6 + 19/4 = 40/24 + 57/24 = 97/24 pounds of nuts.

To find the total amount of nuts Martin buys, we need to add the amount of peanuts and the amount of almonds he buys.

First, we need to convert the mixed numbers to improper fractions. To do this, we multiply the whole number by the denominator and add the numerator. For 2(5)/(6), we multiply 2 by 6 and add 5 to get 10/6. For 4(3)/(4), we multiply 4 by 4 and add 3 to get 19/4.

Next, we need to find a common denominator in order to add the two fractions. The least common denominator for 6 and 4 is 24, so we multiply the numerator and denominator of 10/6 by 4 to get 40/24 and the numerator and denominator of 19/4 by 6 to get 57/24.

Finally, we add the two fractions by adding the numerators and keeping the same denominator: 40/24 + 57/24 = 97/24.

Therefore, Martin buys a total of 97/24 pounds of nuts.

To know more about improper fractions click on below link:

https://brainly.com/question/21449807#

#SPJ11

The simplified expression is (3y-7)/(y^2-35)/3.

To simplify the given expression, we need to find the least common denominator (LCD) of the two fractions. The LCD of (y-5) and (y+7) is (y-5)(y+7).
Next, we need to multiply each fraction by the LCD in order to get the same denominator for both fractions.
(2)/(y-5) * (y+7)/(y+7) + (7)/(y+7) * (y-5)/(y-5)
= (2y+14)/(y^2-35) + (7y-35)/(y^2-35)

Now, we can combine the two fractions since they have the same denominator.
(2y+14+7y-35)/(y^2-35)
= (9y-21)/(y^2-35)
Finally, we can simplify the fraction by factoring out a 3 from the numerator.
= (3)(3y-7)/(y^2-35)
= (3y-7)/(y^2-35)/3
You can read more about least common denominator  at https://brainly.com/question/19249494

#SPJ11

Answer:

Option d: Side OP is congruent to side RS.

To prove that ΔNOP ≅ ΔQRS by ASA, we need to show that:

1. ∠N ≅ ∠Q (given)

2. Side NP ≅ Side QS (given)

3. Side OP ≅ Side RS (additional information needed)

Hence, option d is the correct answer.

The final simplified expression turns out to be "-8x^2 + 6x - 6"

To perform the addition or subtraction and simplify the expression, we need to combine like terms and simplify the expression. First, we need to subtract the second expression from the first one:

x^2 + x − 2 - (9x^2 − 5x + 4)

Next, we need to distribute the negative sign to each term in the second expression:

x^2 + x − 2 - 9x^2 + 5x - 4

Now we can combine like terms:

-8x^2 + 6x - 6

This is the simplified expression. Therefore, the answer is:

-8x^2 + 6x - 6

Know more about simplified expression here:

https://brainly.com/question/18077352

#SPJ11

Answer:120

Step-by-step explanation:

The correct equation for the sentence is: 24 = 31.4n - 8.4.

Twenty-four is equal to 31.4 times a number plus a negative 8.4 multiplied by the proper equation, which is:

24 = 31.4n - 8.4

This formula can be rewritten as follows:

31.4n = 24 + 8.4

And after being distilled:

31.4n = 32.4

Finally, by multiplying both sides by 31.4, we can find the value of n:

n = 32.4/31.4

Thus, the following is the proper equation for the phrase:

24 = 31.4n - 8.4.

To know more about Equation visit:

https://brainly.com/question/29657988

#SPJ1

The mean, median and mode of the prices that the bags of flour are sold at are:

To find the mean, we add up all the prices and divide by the total number of prices:

Mean:

= (1.45 + 1.55 + 1.90 + 1.95 + 1.45) / 5

= 1.66

To find the median, we need to arrange the prices in order from lowest to highest:

1.45, 1.45, 1.55, 1.90, 1.95

There are five prices, so the median is the middle value, which is 1.55.

To find the mode, we need to look for the price that appears most frequently in the data set. In this case, there are two prices that appear twice, 1.45 and 1.55. So there is no unique mode for this data set.

Find out more on mean at https://brainly.com/question/14532771

#SPJ1

The percent increase of the puppy's weight is 25%

A puppy weighed 28 ounces on Monday

It gained 7 ounces in one week

Total weight is = 28 + 7

= 35

The percent increase can be calculated as follows

= 35/28

= 1.25

= 1.25-1

= 0.25 × 100

= 25%

Hence the percent increase of the puppy's weight is 25%

Read more on percent increase here

https://brainly.com/question/20066775

#SPJ1

The area of the isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft will be 8.75ft

Isosceles trapezoid, also known as isosceles trapezium, is defined as a convex quadrilateral, which mainly consists of a line of symmetry that bisects one pair of the opposite side, it is also know that the base angles are of equal measure. It has parallel bases with the legs also being of the equal measure, on the other hand the opposite angles of the isosceles trapezoid are supplementary, which makes it a cyclic quadrilateral.

Area of an Isosceles Trapezoid = [(a+b)h]/2 square units,

when we put these values in the formula, we get:

Area = [(2+5)2.5]/2

Area = (7 × 2.5)/2

Area = 17.5/2

Area = 8.75ft²

therefore, we know that the area of the isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft will be 8.75ft

To learn more about area, click here:

brainly.com/question/27683633

#SPJ1

The side x is 30 miles

The unknown side is 346 yards

The sine rule, also known as the law of sines, is a formula that relates the sides and angles of a non-right triangle (a triangle that does not have a 90-degree angle). The sine rule states that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides of the triangle.

1) Using the sine rule;

x/sin 65 = 32/sin75

xsin75 = 32sin65

x = 32sin65/sin75

x = 30 miles

2) The angle = π/3 = 180/3 = 60 degrees

Using the cosine rule;

c^2 = (200)^2 + (400)^2 - (2 * 200 * 400) cos 60

c^2 = 200,000 - 80,000

c = 346 yards

Learn more about cosine rule:https://brainly.com/question/28716982

#SPJ1

Step-by-step explanation:

To find the x-intercepts of the quadratic function f(x) = x^2 + 6x - 27, we need to set f(x) equal to zero and solve for x.

So we have:

f(x) = 0

x^2 + 6x - 27 = 0

We can solve for x using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Where a = 1, b = 6, and c = -27.

Substituting these values into the quadratic formula, we get:

x = (-6 ± sqrt(6^2 - 4(1)(-27))) / 2(1)

x = (-6 ± sqrt(180)) / 2

x = (-6 ± 6sqrt(5)) / 2

x = -3 ± 3sqrt(5)

So the x-intercepts of the quadratic function are approximately -10.16 and 4.16.

The simplified expression is  [tex]12u^8[/tex].

What is simplification?

Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression: 4 + 6 + 5.

Here the given expression is ,

=> [tex]-2u^5(-6u^3)[/tex]

Now simplifying the expression,

=> [tex]-2\times -6 \times u^5 \times u^3[/tex]

=> 12[tex]u^{5+3}[/tex]

=> [tex]12u^8[/tex]

Hence the simplified expression is  [tex]12u^8[/tex].

To learn more about Simplification refer the below link

https://brainly.com/question/20373987

#SPJ1

The coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).

Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.

here, we have,

From the given graph, triangle ABC have A(1, 3), B(7, 1) and (1, 1).

Triangle ABC is enlarged with a scale factor of -2 and the origin of the Centre to give triangle A'B'C'.

Now, A(1, 3) → -2(1, 3)→A'(-2, -6)

B(7, 1) → -2(7, 1) → B'(-14, -2)

C(1, 1) → -2(1, 1) → C'(-2, -2)

Therefore, the coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).

To learn more about the dilation visit:

brainly.com/question/13176891.

#SPJ1

(3x²+18x)/(7x+21)  is the  simplified answer of  (x+6)/(3x+9)-:(7)/(9x)+42.

To divide the given expressions, we need to first simplify the expressions as much as possible and then use the rule for dividing fractions. The rule for dividing fractions is to multiply the first fraction by the reciprocal of the second fraction.

Simplifying the expressions:

(x+6)/(3x+9) = (x+6)/(3(x+3)) = (x+6)/3(x+3)

(7)/(9x) = 7/9x

Now, using the rule for dividing fractions:

((x+6)/3(x+3)) ÷ (7/9x) = ((x+6)/3(x+3)) * (9x/7)

= (9x(x+6))/(3(x+3)*7)

= (3x(x+6))/(x+3)*7

= (3x²+18x)/(7x+21)

To know more about reciprocal click on below link:

https://brainly.com/question/18634761#

#SPJ11

A $900 item on the sale of 20% discount the

Money off: $180

Amt. of tax: $50.40

Money paid: $770.40

Final price: $770.40

what is discount ?

A discount is a reduction in the price of an item or service, often offered as a promotion or incentive to encourage customers to make a purchase. The discount is usually expressed as a percentage of the original price and represents the amount of money that the customer can save on the purchase.

For example:

Percentage discount: a certain percentage of the original price is deducted from the selling price. For example, a 20% discount on a $100 item would reduce the price to $80.

According to the question:
To calculate the final price of the item with sales tax, we can follow these steps:

Calculate the amount of money off by multiplying the original price by the discount percentage:

Money off = 0.20 x $900 = $180.

Calculate the price after the discount by subtracting the money off from the original price:

Price after discount = $900 - $180 = $720.

Calculate the amount of sales tax by multiplying the price after discount by the tax rate:

Amount of tax = 0.07 x $720 = $50.40.

Calculate the money paid by adding the price after discount and the amount of tax:

Money paid = $720 + $50.40 = $770.40.

Therefore, the final price with 7% sales tax is $770.40.

To know more about discount visit:
https://brainly.com/question/29124624
#SPJ1

The situations that best represent the scenario shown in the graph of the quadratic functions, y = x² and y = x² + 3 include the following:

B. "From x = -2 to x = 0, the average rate of change for both functions is negative."

C. "For the quadratic function, y = x² + 3, the coordinate (2, 7) is a solution to the equation of the function."

D. "The quadratic function, y = x², has an x-intercept at the origin."

In Mathematics, the x-intercept of the graph of any function simply refers to the point at which the graph of a function crosses or touches the x-coordinate (x-axis) and the y-value or value of "y" is equal to zero (0).

In this context, we can logically deduce that the x-intercepts of the graph of the given equation y = x² is at the origin:

x = (0, 0)

For the the coordinate (2, 7), we would evaluate the quadratic function, y = x² + 3 as follows;

7 = 2² + 3

7 = 4 + 3

7 = 7 (True).

By critically observing the graph of both functions, we can logically deduce that their average rate of change is negative from x = -2 to x = 0.

Read more on x-intercept here: https://brainly.com/question/2606550

#SPJ1

Answer: 2

Step-by-step explanation:

Answer and Explanation: The only even number that is not a composite number is 2. All even numbers are defined as numbers that are divisible by, or have a divisor of, 2.

Answer:

The answer to your question is, 2

Step-by-step explanation:

The reason why that is our answer is because, 2 can either be multiplied and divided but 2 NUMBERS.

Such as the following:

2 / 1 = 2.

1 x 2 = 2.

What is a composite number?

A composite number are numbers that have more than two factors like the number 2:

2 / 1 = 2.

1 x 2 = 2.

Thus the answer to your problem is, 2

All sides of the given rhombus measure 5 cm respectively.

A rhombus is a quadrilateral with four equal-length sides in planar Euclidean geometry.

The term "equilateral quadrilateral" refers to a quadrilateral whose sides all have equal lengths.

Quadrilaterals having four congruent sides are called squares.

Squares are rhombuses by definition since they are quadrilaterals with four congruent sides.

So, consider the following values: AC = 8 cm, BD = 6 cm, OD = OB = 3 cm, and OA = OC = 4 cm.

Now,

OB² + OC² = BC²

3²+ 4² = BC²

9 + 16 = BC²

25 = BC²

5 = BC

A rhombus has an equal number of sides.

Therefore, all sides of the given rhombus measure 5 cm respectively.

Know more about the rhombus here:

https://brainly.com/question/88523

#SPJ9

Answer:

A) $8

B) 4+2x

Step-by-step explanation:

If it costs $2 for every mile driven, and Huoa rides for 2 miles, then you multiply 2×2=4. Then you add another $4 for the $4 the driver charges for just getting in, so 4+4=8

Another way to do that is to use the equation that's the answer for part B. As you don't know the number of miles, you substitute that unknown number for x.

4+2x

The x represents the number of miles

The 2 represents the amount charged for each mile

The 4 represents the flat fee for getting into the car

If you plug 2 (for the 2 miles driven) into the equation you get:

4+2(2)

4+4

8

The solutions of the given functions are:

f(g(x)) = x+10√x+26 and g(f(x)) = √(x²+6)

f(x) = x²+1, g(x) = √x+5 are given.

To find f(g(x)), we need to substitute g(x) into the function f(x):

f(g(x)) = f(√x+5) = (√x+5)²+1 = x+10√x+26

To find g(f(x)), we need to substitute f(x) into the function g(x):

g(f(x)) = g(x²+1) = √(x²+1+5) = √(x²+6)

Therefore, the simplified answers are:

f(g(x)) = x+10√x+26

g(f(x)) = √(x²+6)

Note: It is important to include the parentheses when substituting one function into another to ensure the correct order of operations.

For more such questions on Functions.

https://brainly.com/question/12431044#

#SPJ11

Answer:

335.6511628

Step-by-step explanation:

Ux258-336=168

U-336=0.6511628

U=336.6511628

Answer:

Step-by-step explanation:

You take 528- divided by 2000- and then you get your answer.

Answer:

1.  c

2. b

3.  a

Step-by-step explanation:

Surface Area of a cylinder = 2πrh + 2πr²

1.  2(3.14)(5)(11) + 2(3.14)(5)² = 502.4

2. 2(3.14)(6)(10) + 2(3.14)(6)² = 602.88

3.  2(3.14)(2)(12) + 2(3.14)(2)² = 175.84

The statement which describes the decimal equivalent of 5/16 is option(a), It is a decimal that terminates after 4 decimal places.

To convert a fraction to a decimal, we need to divide the numerator by the denominator.

In this case, we want to find the decimal equivalent of the fraction 5/16,

⇒ 5 ÷ 16 = 0.3125

So, the decimal equivalent of 5/16 is 0.3125.

The option(a) states that, It is a decimal that terminates after 4 decimal places.

This statement is correct, because the decimal equivalent of 5/16 terminates after 4 decimal places.

Therefore, the correct statement is option is (a).

Learn more about Decimal Equivalent here

https://brainly.com/question/9511032

#SPJ4

The given question is incomplete, the complete question is

What statement describes the decimal equivalent of 5/16?

(a) It is a decimal that terminates after 4 decimal places.

(b) It is a decimal with a repeating digit of 5.

(c) It is a decimal that terminates after 3 decimal places.

(d) It is a decimal with repeating digits of 6.

It wοuId take 3 hοurs fοr 96 cars tο pass thrοugh the intersectiοn at the same rate as 32 cars in 1 hοur.

We can use the cοncept οf prοpοrtiοnaIity tο sοIve this prοbIem. The number οf cars passing thrοugh the intersectiοn is directIy prοpοrtiοnaI tο the amοunt οf time taken. This means that if we dοubIe the number οf cars passing thrοugh, we wiII aIsο dοubIe the amοunt οf time taken.

Let x be the amοunt οf time it takes fοr 96 cars tο pass thrοugh the intersectiοn. Then, we can set up the fοIIοwing prοpοrtiοn:

32 cars / 1 hοur = 96 cars / x hοurs

Tο sοIve fοr x, we can crοss-muItipIy and simpIify:

32x = 96

x = 3

Therefοre, it wοuId take 3 hοurs fοr 96 cars tο pass thrοugh the intersectiοn at the same rate as 32 cars in 1 hοur.

To know more about intersection:

brainly.com/question/14217061

#SPJ1