solve the following problem

Answer:

Step-by-step explanation:

I think there is a typo in the question

Qn : let p(n) be  [tex]\sum\limits^n_{i = 1}{i2^i} = 2 + (n-1)2^{n+1} \;\;\;\;\;\;:n\geq 1[/tex]

When n = 1:

LHS: 1(2¹) = 2

RHS: 2 + (1 - 1)(2¹ ⁺ ¹) = 2

LHS = RHS

⇒ p(n) holds for n = 1

Let us assume that the proof holds for p(n): n = x

ie.

[tex]p(x): \sum\limits^x_{i = 1}{i2^i} = 2 + (x-1)2^{x+1}[/tex]

To prove that the proof holds for n = x+1

ie   [tex]p(x+1): \sum\limits^{x+1}_{i = 1}{i2^i} = 2 + (x)2^{x+2}[/tex]

Consider LHS

[tex]\sum\limits^{x+1}_{i = 1}{i2^i}\\\\= \sum\limits^{x}_{i = 1}{i2^i}+\sum\limits^{x+1}_{i = x+1}{i2^i}\\\\= p(x) + (x+1)2^{x+1}\\\\= 2 + (x-1)2^{x+1} + (x+1)2^{x+1}\\\\= 2 + 2^{x+1} (x-1 + x+1)\\\\= 2 + 2^{x+1} (2x)\\\\= 2 + (x)2^{x+2}\\\\= RHS[/tex]

[tex]\begin{gathered} \; \color{pink}{\frak{\qquad \: x - \dfrac{x - 1}{2} = 1 - \dfrac{x - 2}{3}}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{\dfrac{(2 \times x) - x - 1}{2} = \dfrac{(3 \times 1) - x - 2}{3}}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{\dfrac{2x - x - 1}{2} = \dfrac{3 - x - 2}{3}}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{\dfrac{x - 1}{2} = \dfrac{1 - x}{3}}} \\ \end{gathered}[/tex]

Cross Multiplying,,

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{3(x - 1) = 2(1 - x)}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{3x - 3 = 2 - 2x}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{3x + 2x = 2 + 3}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{5x = 5}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink}{\hookrightarrow \: \frak{x = \dfrac{\cancel{ \: 5}}{\cancel{ \: 5}}}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{cyan}{\hookrightarrow \: \underline{\boxed{\frak{x = 1}}}} \: \pmb{\bigstar} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \qquad{\qquad{\pmb{━━━━━ღ◆ღ━━━━━}}} \end{gathered}[/tex]

According to the information we can infer that the period of the graph is 8.

To determine the period of the graph we have to consider that the period of a grah is the distance between rigdes. So, in this case we have to count what is the difference between each rigde.

In this case, the distance between rigdes is 8 units because the first is located in the line 1 an the second is located in the line 9. So we can conclude that the period of the graph is 8.

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The amount of carpet needed for the living room is three times greater than the amount of carpet needed for the bedroom.

To calculate the amount of carpet needed for each room, we need to consider the dimensions of the rooms. Let's assume the length of the bedroom is x units.
Since the living room is two times as long as the bedroom, its length will be 2x units.
Similarly, the width of the living room is one and a half times as wide as the bedroom. So, the width of the living room will be 1.5x units.
To find the area of each room, we multiply the length by the width.
The area of the bedroom is x units * x units = x^2 square units.
The area of the living room is 2x units * 1.5x units = 3x^2 square units.
The amount of carpet needed is directly proportional to the area of the room. Therefore, the amount of carpet needed for the living room is 3x^2 square units, while the amount needed for the bedroom is x^2 square units.
To find the difference, we divide the area of the living room by the area of the bedroom: 3x^2 / x^2 = 3.
Thus, the amount of carpet needed for the living room is three times greater than the amount needed for the bedroom.

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The period of oscillation of the wave is 8 seconds

The time taken for an oscillating particle to complete one cycle of oscillation is known as the Period of oscillating particle.

Period is measured in seconds and it is the inverse of frequency of the oscillation. This means that T = 1/f

where f is the frequency and T is the period.

An oscillation can also be called cycle or Revolution or vibration.

From the graph, the oscillating wave made a complete oscillation at 8 seconds.

Therefore the period of oscillation of the wave is 8 seconds.

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The solutions for x are approximately x ≈ 7.888 and x ≈ -4.263.

To solve the equation, let's simplify the equation and then isolate the variable x.One of the properties of triangle Pythagorous theorem

The given equation is: 27^2 + (4x - 4)^2 = 36^2.

First, evaluate the exponents: 27^2 = 729 and 4^2 = 16.

Substituting these values back into the equation, we have: 729 + (4x - 4)^2 = 1296.

Next, simplify the equation further: 729 + (4x - 4)(4x - 4) = 1296.

Expanding the squared term: 729 + (16x^2 - 32x + 16) = 1296.

Combine like terms: 16x^2 - 32x + 729 + 16 = 1296.

Simplify further: 16x^2 - 32x + 745 = 1296.

Rearrange the equation: 16x^2 - 32x + 745 - 1296 = 0.

Combine like terms: 16x^2 - 32x - 551 = 0.

To solve the quadratic equation, we can either factor it or use the quadratic formula. However, this equation does not factor easily, so we'll use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a).

In this case, a = 16, b = -32, and c = -551.

Substituting these values into the quadratic formula:

x = (-(-32) ± √((-32)^2 - 4(16)(-551))) / (2(16)).

Simplifying further:

x = (32 ± √(1024 + 35328)) / 32.

x = (32 ± √(36352)) / 32.

x = (32 ± 190.413) / 32.

Therefore, the two possible solutions for x are:

x1 = (32 + 190.413) / 32 ≈ 7.888.

x2 = (32 - 190.413) / 32 ≈ -4.263.

Therefore, the solutions for x are approximately x ≈ 7.888 and x ≈ -4.263.

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f'(x) = 0 * x^(-2) + (-2/x^3) = -2/x^3.

So, the derivative of f(x) = 2/x^2 is f'(x) = -2/x^3.

To find f'(x) for the function f(x) = x + 2, we can use the power rule for derivatives.

The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = n*x^(n-1).

In this case, the function f(x) = x + 2 can be written as f(x) = x^1 + 2.

Applying the power rule, we differentiate each term separately:

f'(x) = d/dx (x^1) + d/dx (2)

The derivative of x^1 is 1x^(1-1) = 1x^0 = 1.

The derivative of a constant term like 2 is 0, as the derivative of a constant is always 0.

Therefore, f'(x) = 1 + 0 = 1.

So, the derivative of f(x) = x + 2 is f'(x) = 1.

To find f'(x) for the function f(x) = 2/x^2, we can use the power rule and the constant multiple rule.

The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = n*x^(n-1).

The constant multiple rule states that if we have a function of the form f(x) = cg(x), where c is a constant, then the derivative is given by f'(x) = cg'(x), where g'(x) is the derivative of g(x).

In this case, the function f(x) = 2/x^2 can be written as f(x) = 2 * x^(-2).

Applying the power rule and the constant multiple rule, we differentiate each term separately:

f'(x) = d/dx (2 * x^(-2))

Applying the constant multiple rule, the derivative of 2 is 0, as it is a constant term.

Applying the power rule, the derivative of x^(-2) is (-2) * x^(-2-1) = (-2) * x^(-3) = -2/x^3.

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Answer:To determine the range is the same as to determine which numbers appear as the second number (the y-value) in an ordered pair that is part of the graph. Here are some examples: y ≥ x2 + 3. graph {y >= x^2+3 [-11.6, 13.72, 0.15, 12.81]} Although it is not 100% certain from just the graph, this graph does get wider and wider.

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

the x- axis is a horizontal line. A line perpendicular to it will be a vertical line, parallel to the y- axis with equation

x = c ( c is the value of the x- coordinates the line passes through )

the only equation fitting this description from the list is

x = 3

Answer:

  (c)  31 units

Step-by-step explanation:

Given quadrilateral ABCD has AD║BC, with AD=3x+7 and BC=5x-9, you want to know the length of AD for the quadrilateral to be a parallelogram.

Opposite sides of a parallelogram are congruent, so for ABCD to be a parallelogram, we must have ...

  BC = AD

  5x -9 = 3x +7

  2x = 16

  x = 8

  AD = 3x +7 = 3(8) +7 = 31

The length of AD must be 31 units if ABCD is to be a parallelogram.

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Answer: D

Step-by-step explanation:

we will multiply

4/7* 1/8

we will get 1/14

Answer:

Step-by-step explanation:
After reducing 4/7 of 1/8 to the lowest terms we get 1/14. Thus, option C is the answer.

For reducing a number to its lowest terms, first, we need to simplify the numbers to a more significant number, i.e. 4/7 of 1/8 means 4/7*1/8=4/56.

Now we know that 56 is a multiple of 4 when multiplied by 14. Hence, when we get it in the lowest term we would get 1/14

The probability of the four events are: Event 1: 1/84Event 2: 3/14Event 3: 15/28 Event 4: 5/21

The total number of drinks = 9The number of sweet teas = 3The number of unsweetened teas = 6If you select 3 drinks at random, the following events can take place:

Event 1: All three drinks are sweet teas. The probability of event 1 = (Number of ways in which all three drinks can be sweet teas) / (Number of ways to select 3 drinks)The number of ways in which all three drinks can be sweet teas = 3C3 = 1 (because all three sweet teas are already fixed)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84Therefore, the probability of event 1 = 1/84 = 1/84

Event 2: Exactly two drinks are sweet teas. The probability of event 2 = (Number of ways in which two drinks are sweet teas and one is an unsweetened tea) / (Number of ways to select 3 drinks)The number of ways in which two drinks are sweet teas and one is an unsweetened tea = (3C2 × 6C1) = 18 (because you can choose 2 sweet teas from 3 and 1 unsweetened tea from 6)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84Therefore, the probability of event 2 = 18/84 = 3/14

Event 3: Exactly one drink is a sweet tea. The probability of event 3 = (Number of ways in which one drink is a sweet tea and the other two are unsweetened teas) / (Number of ways to select 3 drinks)The number of ways in which one drink is a sweet tea and the other two are unsweetened teas = (3C1 × 6C2) = 45 (because you can choose 1 sweet tea from 3 and 2 unsweetened teas from 6)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84Therefore, the probability of event 3 = 45/84 = 15/28

Event 4: All three drinks are unsweetened teas. The probability of event 4 = (Number of ways in which all three drinks can be unsweetened teas) / (Number of ways to select 3 drinks)The number of ways in which all three drinks can be unsweetened teas = 6C3 = 20 (because you can choose 3 unsweetened teas from 6)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84 Therefore, the probability of event 4 = 20/84 = 5/21

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Answer:

y = [tex]\frac{-5}{4}[/tex]x - 30

Step-by-step explanation:

we will use the x = -40 and the y = 20 from the point given (-40,20).  The perpendicular slope would be the opposite reciprocal from the slope given.  The slope given is 4/5.  The reciprocal is 5/4 and the opposite would be -5/4

y = mx + b  Plug in what we know and solve for b

20 = [tex]\frac{-5}{4}[/tex] ( -40) + b

20 = [tex]\frac{-5}{4}[/tex] ·[tex]\frac{-40}{1}[/tex] + b

20 = [tex]\frac{200}{4}[/tex] + b

20 = 50 + b  Subtract 50 from both sides

20 - 50 = 50 - 50 + b

-30 = b

To write the equation we need the slope (m) ([tex]\frac{-5}{4}[/tex]) and the y-intercept (b) (-30)

y = mx + b

y = [tex]\frac{-5}{4}[/tex]x - 30

Helping in the name of Jesus.

Given statement solution is :- "F(g(x)) g(f(x))": The function F(x) is composed with g(x), and the result is further composed with g(f(x)).

"f(g(x)) g(f(x))": The function g(x) is composed with f(x), and the result is further composed with g(f(x)).

The expressions "F(g(x)) g(f(x))" and "f(g(x)) g(f(x))" are compositions of functions. The answer to the question posed would depend on the specific functions F(x) and g(x), as well as f(x) and g(x) in the second expression.

To provide a brief response:

For the expression "F(g(x)) g(f(x))": The function F(x) is composed with g(x), and the result is further composed with g(f(x)). The order of composition is F(g(x)) first, followed by g(f(x)).

For the expression "f(g(x)) g(f(x))": The function g(x) is composed with f(x), and the result is further composed with g(f(x)). The order of composition is f(g(x)) first, followed by g(f(x)).

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Answer:

It looks to be -4, -2, 1, and 3

Answer:

418

Step-by-step explanation:

-4.0(5+100)+80(5)+60

We will consider the '+' outside the bracket as the dividers.

Therefore,

=-0.4(105)+400+60

=-42+400+60

=358+60

=418

Answer:

Step-by-step explanation:

try (3,-1)

Answer:

Step-by-step explanation:

A rhombus has all equal sides and two parallel sides.

Notice how Point C is 2 units to left and 1 unit below Point D>

Thus, our missing point, A, should be 2 units to left of B and 1 unit below it.

We therefore get (1,3)

So the point is (1,3)

Answer:

Step-by-step explanation: f'(x)=0. Explanation: f'(x)=8x+2=0. critical value =−14. hope that helped.

Answer:

Pamela is 30 years old.

Step-by-step explanation:

First equation:  

Since Pamela is 3 times older than Jakob, our first equation is given by:

P = 3J

Second Equation:

Since Pamela will be twice as old as Jakob in 10 years, our second equation is given by:

P = 2J + 10

Method to solve:  Substitution:

We can solve with substitution by isolating J in the second equation.  This will allow us to substitute it for J in the second equation and find P, Pamela's age:

Isolating J:

Step 1:  Divide both sides by 3

(P = 3J) / 3

P/3 = J

Substituting P/3 = J for J in P = 2J + 10:

P = 2(P/3) + 10

Step 1:  Distribute the 2 to P/3:

P = 2/3P + 10

Step 2:  Multiply both sides by 3 to clear the fraction:

(P = 2/3P + 10) * 3

3P = 2P + 30

Step 3:  Subtract 2P from both sides:

(3P = 2P + 30) - 2P

P = 30

Step 4:  Divide both sides by 2 to find P, Pamela's age:

(2P = 30) / 2

P = 30

Thus, Pamela is 30 years old.

Optional Steps to check the validity of our answer:

Plugging in 30 for P in P = 3J:

Step 1:  Divide both sides by 3:

(30 = 3J) / 3

10 = J

Thus, Jakob is 10 years old.

Checking the validity of answers with verbal statements:

Since 30 (i.e., Pamela's age) is indeed 3 times 10 (i.e., Jakob's age), this satisfies the first statement.

In 10 years, Pamela will be 40 as 30 + 10 = 40.

In 10 years, Jacob will be 20 as 10 + 10 = 20.

Since 40 (i.e., Pamela's age in 10 years) is indeed twice 20 (i.e., Jakob's age in 10 years), this satisfies the second statement.

Thus, our answer for Pamela's age is correct.

Answer:

45

Step-by-step explanation:

Do the operation in parentheses first.

(45)(4 - 3) = (45)(1) = 45

Answer:

Step-by-step explanation:

The answer is 45.

Given,

[tex](45)(4-3)\\=(45)(1)\\=45[/tex]Using subtraction and multiplication rule.

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Answer:

a) [tex]\frac{x^2-4}{(x + 7)(x - 4)(x+2)}[/tex]

b)[tex]\frac{x^2-4x}{(x + 7)(x + 2)(x-4)}[/tex]

Step-by-step explanation:

x² + 3x - 28

= x² + 7x - 4x -28

= x(x + 7) - 4(x + 7)

=

= x² + 7x + 2x + 14

= x(x + 7) + 2(x + 7)

= (x + 7)(x + 2)

LCM of x² + 3x - 28 and x² + 9x + 14 is

(x + 7)(x - 4)(x + 2)

We can write:

[tex]\frac{x-2}{x^2 + 3x - 28}\\ \\= \frac{x-2}{(x + 7)(x - 4)}\\\\= \frac{x-2}{(x + 7)(x - 4)} *\frac{x+2}{x+2} \\\\=\frac{(x-2)(x+2)}{(x + 7)(x - 4)(x+2)} \\\\=\frac{x^2-2^2}{(x + 7)(x - 4)(x+2)} \\\\=\frac{x^2-4}{(x + 7)(x - 4)(x+2)}[/tex]

and

[tex]\frac{x}{x^2 + 9x + 14 }\\ \\= \frac{x}{(x + 7)(x + 2)}\\\\= \frac{x}{(x + 7)(x + 2)} *\frac{x-4}{x-4} \\\\= \frac{x(x-4)}{(x + 7)(x + 2)(x-4)} \\\\= \frac{x^2-4x}{(x + 7)(x + 2)(x-4)}[/tex]

Answer:the space is 7 in the text below

Step-by-step explanation:

Answer:

-12

Step-by-step explanation:

3/5 * (1 + sqrt(16))^2 - (5 - 2)^3 =

= 3/5 * (1 + 4)^2 - (3)^3

= 3/5 * (5)^2 - 27

= 3/5 * 25 - 27

= 15 - 27

= -12

Answer:

i dont know

Step-by-step explanation:

give proper diagram please

Answer:

65.56°

Step-by-step explanation:

Trigonometric ratios:

        Opposite side of x° is 11 cm and adjacent side of x° is 5 cm.

To find x°, we have to use Tan ratio.

           [tex]\boxed{\bf Tan \ x = \dfrac{opposite \ side \ of \ \angle x }{adjacent \ side \ of \angle x}}[/tex]

                       [tex]\sf = \dfrac{11}{5}\\\\ = 2.2[/tex]

                    [tex]\sf x = tan^{-1} \ 2.2\\\\ x = 65.56[/tex]

Answer:

a. The probability that this whole shipment will be accepted is the probability that none or only one of the 42 randomly selected tablets is defective.

b. The company will accept approximately 36.8% of the shipments and will reject approximately 63.2% of the shipments, so many shipments will be rejected.

Step-by-step explanation:

Answer:

To determine the probability that the whole shipment will be accepted, we need to calculate the probability that at most one battery does not meet specifications.

Given that 1% of the batteries do not meet specifications, it means that out of 5000 batteries, 0.01 * 5000 = 50 batteries do not meet specifications.

Now, using the binomial probability formula, we can calculate the probability of at most one battery not meeting specifications:

P(X ≤ 1) = P(X = 0) + P(X = 1)

Where X follows a binomial distribution with parameters n = 60 (sample size) and p = 50/5000 = 0.01 (probability of a battery not meeting specifications).

P(X = 0) = (60 choose 0) * (0.01)^0 * (1 - 0.01)^(60 - 0)

P(X = 1) = (60 choose 1) * (0.01)^1 * (1 - 0.01)^(60 - 1)

Calculating these probabilities:

P(X = 0) ≈ 0.301

P(X = 1) ≈ 0.401

Therefore,

P(X ≤ 1) = P(X = 0) + P(X = 1) ≈ 0.301 + 0.401 ≈ 0.702

a. The probability that this whole shipment will be accepted is approximately 0.702.

b. The company will accept approximately 70.2% (0.702 * 100) of the shipments and will reject approximately 29.8% (100 - 70.2) of the shipments.

Answer:

To get the price of 1 each

$80.50÷7=

$11.50

To get the price of 10

$11.50×10

=$115

Answer:

Step-by-step explanation:

To find the ratio of correct to incorrect answers on the test, we need to determine the number of correct and incorrect answers.

If you got 95% of the questions correct, it means you got 95% of 140 questions correct.

Number of correct answers = 95% of 140 = (95/100) * 140 = 133

To find the number of incorrect answers, subtract the number of correct answers from the total number of questions:

Number of incorrect answers = Total number of questions - Number of correct answers = 140 - 133 = 7

Therefore, the ratio of correct to incorrect answers on the test is 133:7, which is already in its lowest terms and cannot be simplified further.

Final result:

Of the 140 questions, you properly answered 133 of them, while 7 of them were incorrect. You have a 133:7 chance of winning because you provided the right answers.

Explanation:

We need to know how many questions were correctly and wrongly answered before we can fix this issue. You correctly answered 140 out of 140 questions, or 95 percent of them, giving you a total of 133 valid answers.

Thus, 133 correct answers out of 140 total questions equals 7 incorrect answers. As a result, the ratio of correct to incorrect answers is already very low at 133:7.

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Answer:

1711

Step-by-step explanation:

[tex]4\cdot(14+8)^2-9\cdot(9-4)^2\\=4\cdot(22)^2-9\cdot(5)^2\\=4(484)-9(25)\\=1936-225\\=1711[/tex]

Make sure to follow order of operations!