Part b assume the statement is true for n = k. prove that it must be true for n = k + 1, therefore proving it true for all natural numbers, n. t hint: since the total number of dots increases by n each time, prove that d (k) + (k + 1) = d(k+1).

Answers

Answer 1

The he statement d(k) + (k + 1) = d(k+1) is true for all natural numbers, n, by mathematical induction.

To prove that the statement is true for all natural numbers, n, we can use mathematical induction.

The statement we want to prove is that d(k) + (k + 1) = d(k + 1), where d(n) represents the total number of dots in a pattern of n squares.

Base Case (n = 1):

First, let's prove the statement for the base case, n = 1.

For n = 1:

d(1) + (1 + 1) = d(1) + 2

Now, consider a single square with 1 dot.

In this case, d(1) = 1.

So, we have:

1 + 2 = 3

Now, let's consider a pattern of 2 squares (n = 2).

The first square has 1 dot, and the second square has 2 dots. So, d(2) = 1 + 2 = 3.

So, the statement is true for n = 1.

Inductive Hypothesis (Assume true for n = k):

Now, assume that the statement is true for some arbitrary natural number k.

That is, assume that: d(k) + (k + 1) = d(k + 1)

Inductive Step (Prove true for n = k + 1):

To prove that the statement is true for n = k + 1.

d(k + 1) + (k + 2) = d(k + 2)

Now, consider a pattern of (k + 1) squares.

By the inductive hypothesis, assume that the statement is true for k squares:

d(k) + (k + 1) = d(k + 1)

Now, let's add one more square to the pattern.

This square will have (k + 2) dots.

So, the total number of dots in the pattern of (k + 1) squares plus the (k + 2) dots in the additional square is:

d(k + 1) + (k + 2)

And by the inductive hypothesis, d(k) + (k + 1) = d(k + 1).

Therefore:

d(k + 1) + (k + 2) = (d(k) + (k + 1)) + (k + 2) = d(k) + (k + 1) + (k + 2)

Now, simplify:

d(k + 1) + (k + 2) = d(k + 1) + (k + 1 + 1)

So, it is shown that for n = k + 1, d(k + 1) + (k + 2) = d(k + 1) + (k + 1) + 1.

Since it is assumed the statement to be true for k and proved it for k + 1, it is shown that the statement is true for all natural numbers, n, by mathematical induction.

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Related Questions

At what values of x, does f(x) = 0?
a. 0
b. -1
c. -8
d. 4.2
e. 4
f. 2​

Answers

Answer:

B E F

Step-by-step explanation:

as the graphic shows, f(x) = 0 means that the y value is 0. and that means these are the points, where the curve intersects the x-axis.

they are already marked in the graphic :

for x = -1, 2, 4

Subtract 6x^{2}-7x-2 from 6x^{2}-2x+6

Answers

Hello!

Let's subtract!

⇒ 6x² - 2x + 6 - (6x² - 7x - 2)

⇒ 6x² - 2x + 6 - 6x² + 7x + 2

5x + 8

∴ The difference will be equal to 5x + 8.

Answer:

Your answer is 5x+8

Step-by-step explanation:

= 6x^2-2x+6 - (6x^2-7x-2)

= 6x^2-2x+6 - 6x^2+7x+2

= 6x^2- 6x^2-2x+7x+6+2

=5x+8

Hope its helpful!

What is the scale factor from ABC to ADEF?
5
A 3 C
D
15
OA. 2
OB. 3
OC. 3
OD. 1/1/20
E
12

Answers

Answer:

OB

Step-by-step explanation:

no clear explanation for this answer. a picture to demonstrate how i archived it will be shared

The scale factor from triangle ABC to DEF would be,

⇒ 1 / 3

What is mean by Triangle?

A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

We have to given that;

Two triangle ABC and DEF is shown in figure.

Hence, We can formulate;

The scale factor from triangle ABC to DEF would be,

⇒ AC / DF

⇒ 3 / 9

⇒ 1 / 3

Thus, The scale factor from triangle ABC to DEF would be,

⇒ 1 / 3

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4/15 divided by 1 1/3 + 0.4

Answers

Answer: 2/13

Step-by-step explanation:

To solve this word problem expression, we can first convert the mixed number to an improper fraction so it's easier to work with.

1 1/3 = 4/3

Adding 4/3 to 0.4, we get 5.2/3.

4/15 divided by 5.2/3 is 4/26, or 2/13

Please help! Which is the solution set of the following inequality?

Answers

Answer:

x<3

Step-by-step explanation:

3x - 4 < 17 - 4x

So, the -4 shall cross to the other side and replace the -4x which will now take the place of where the -4 used to be and the signs of both digits will change to a positive (+)

Remember the inequality sign (<) shall not change

so now the equation shall appear like this;

3x + 4x < 17 + 4

add both sides and it will appear like this;

7x < 21

simplify both sides by 7 and you'll get

x < 3

HELP ASAP NO ROCKY!!!!!!!!

Answers

Answer:

○ The distance between a and b on the number line is 0.47 units.

Explanation:

The first statement is not true because the distance between a and b on the number line is not 0.47.

Distance between a and b = 5.25 - (-5.72)

                                             = 10.97

• The second statement is true because the sum of a and b is negative:

Sum = a + b

        = -5.72 + 5.25

        = -0.47

• The third statement is true because the quotient of a and -b is:

a ÷ -b

⇒ -5.72 ÷ -5.25

1

And the quotient of -a and b is:

-a ÷ b

⇒ -(-5.72) ÷ 5.25

1

This shows that the quotients are the same.

• The fourth statement is true because the product of a and -b:

a × -b

⇒ -5.72 × -(5.25)

⇒ 30.03 ,

is positive.

Given the following information, determine which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.

Answers

1. Lines a and b are parallel by the converse of alternate interior angles theorem

2. l║m by the converse of consecutive interior angles theorem.

What is the Converse of Alternate Interior Angles Theorem?

Given that the interior angles, ∠3 ≅ ∠7, according to the converse of alternate interior angles theorem, the lines they are found on, lines a and b would be parallel.

1. Lines a and b are parallel.

What is the Converse of Consecutive Interior Angles Theorem?

Given that the interior angles, m∠5 + m∠12 = 180, according to the converse of consecutive interior angles theorem, the lines they are found on, lines l and m would be parallel.

2. Lines l and m are parallel.

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A bird sits at the top of a 24 ft tall tree. A cat, 12 ft away from the base of the tree, sees the bird. To the nearest degree, what is the angle of elevation from the cat to the bird?

Answers

Answer:

https://brainly.com/question/14172055?referrer=searchResults

Step-by-step explanation:

Your question has already been answered.

Here is the link to the answer:

https://brainly.com/question/14172055?referrer=searchResults

Hope this helps!

If not, I am sorry.

help aaahhhh
x =x=x, equals

^\circ



degrees

Answers

Using supplementary angles, it is found that the measure of angle x is of 65º.

What are supplementary angles?

Angles are called supplementary if the sum of their measures is of 180º.

In this problem, angles x, 25, and 90º are supplementary, hence:

x + 25 + 90 = 180

x = 180 - (25 + 90)

x = 65º.

The measure of angle x is of 65º.

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Which statistic is used to determine if there is a significant difference between three or more sets of related scores

Answers

The t-test is a type of inference statistic used to determine if there is a significant difference between the means of two groups that may be related to a particular characteristic. The t-test is one of many tests used to test statistical hypotheses.

Inference statistics have two main purposes. Estimates for population groups (for example, the average SAT score for all 11th graders in the United States). Hypothesis testing to draw inferences about the population

One-way ANOVA is a common method for comparing three or more group means. The usual goal is to determine if the mean (or median) of at least one group is different from the other. Subsequent multiple comparison tests are often used to determine where differences occur.

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Jarom is working for a landscaping company and is responsible for building the stone border around a rectangular pond. The pond measures 6 meters by 15 meters. The manager told Jarom that the budget only allowed enough stone for 46 square meters and to make the stone border as wide as possible. How wide can the border be

Answers

The width is 1 meter.

Given,

Jarom is working for a landscaping company and is responsible for building the stone border around a rectangular pond.The pond measures 6 meters by 15 meters. The manager told Jarom that the budget only allowed enough stone for 46 square meters

Let the border be width  = x

then the area of the border will be

2(6x) + 2(15x) +4x2 = 46

= 12x +30x+ 4x2 = 46

= 4x2 + 42x - 46 = 0

=> x =( -42 +- √42² +4*4*46)/8

x =1 , x = -11.5

since the width is 1 meter.

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Sarah received scores of 85, 92 and 78 on her first three exams of the grading period. What is the lowest score Sarah can earn on the next exam for the exam average to be at least a 87 for the grading period.

Answers

The lowest score could be 87

Use the table to find the products of the two polynomials. Write your answer in
descending order. B) (x2 + x – 2)(4x2 – 8x)

Answers

The products of the two polynomials (x² + x – 2) and (4x² – 8x) is 4x⁴ - 4x³ - 16x² + 16x

What is an equation?

An equation is an expression that shows the relationship between two or more variables and numbers.

The products of the two polynomials (x² + x – 2) and (4x² – 8x) is:

= 4x⁴ - 8x³ + 4x³ - 8x² - 8x² + 16x

= 4x⁴ - 4x³ - 16x² + 16x

The products of the two polynomials (x² + x – 2) and (4x² – 8x) is 4x⁴ - 4x³ - 16x² + 16x

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someone please please help me with this !!

Answers

9) The x-intercepts are when y=0. From the factored form, we know the x-intercepts are (-1,0) and (3,0).

10) The y-intercept is when x=0. From the standard form, we know the y-intercept is (0, -3).

Question 2 of 25
f(x) = 6x + 4. Find the inverse of f(x).
O A. f¹(x) = 4 − 6x
Ỏ B. f-1(2) = *
O c. f¹(x) = -6
OD. f¹(x) = 6x − 4


I NEED YA HELP BRO.!

Answers

The inverse function of f(x) = 6x + 4 is f^-1(x) = (x - 4)/6

How to determine the inverse function?

The function is given as:

f(x) = 6x + 4

Rewrite as:

y = 6x + 4

Swap x and y

x = 6y + 4

Subtract 4 from both sides

6y = x - 4

Divide through by 6

y = (x - 4)/6

Rewrite as:

f^-1(x) = (x - 4)/6

Hence, the inverse function of f(x) = 6x + 4 is f^-1(x) = (x - 4)/6

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Which phrase best describes the translation from the graph y = (x – 5)2 + 7 to the graph of y = (x + 1)2 – 2?

Answers

The parent. function shifted up by 6 units to produce x + 1 and to the left by 9 units

What is translation?

This is a way of changing the position of an object on an xy-plane.

Given the parent function expressed as  y = (x – 5)^2 + 7

From the resulting image after translation, we can see that the parent. function shifted up by 6 units to produce x + 1 and to the left by 9 units

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Which of the following is a fifth root of the given complex number?

Answers

Answer:

D

Step-by-step explanation:

This result is true by De Moivre's theorem.

Please help ill give whoever answers the best the brainliest answer :)))

Answers

Answer:

4th option

Step-by-step explanation:

L = larger softball and S = smaller softball ,then

L + S = 80 → (1) ← equation based on number of softballs ordered

2.75L + 3.25S = 245 → (2) ← equation based on cost

subtract S from both sides in (1)

L = 80 - S → (3)

substitute L = 80 - S into (2)

2.75(80 - S) + 3.25S = 245 ← distribute parenthesis and simplify left side

220 - 2.75S + 3.25S = 245

220 + 0.5S = 245 ( subtract 220 from both sides )

0.5S = 25 ( divide both sides by 0.5 )

S = 50

substitute S = 50 into (3)

L = 80 - 50 = 30

50 small softballs and 30 large softballs were ordered

Answer:

the 4th answer option

Step-by-step explanation:

L, S are the number of balls of that size.

so, L + S = 80. as we order a total of 80 balls.

then each large ball costs $2.75. and each small ball costs $3.25.

so, the total price is

2.75L + 3.25S = 245

as we order balls in total for $245.

so, this set of 2 equations with 2 variables allows us to find the solution.

one defines the total number of balls, and one defines the total amount of money spent by using the individual prices for each type of ball.

the advantage of variables is that we can define the rising and dependencies before we have to calculate any of them specifically.

and once we have all of them defined, then we can start solving the numbers.

The cost of advertising for a radio station is £50+£5 x broadcasting time in minutes use the graph to show the cost of advertising for broadcasting up to 10 minutes of the advert

Answers

A function assigns the values. The cost of advertising for 10 minutes will be equal to £100.

What is a Function?

A function assigns the value of each element of one set to the other specific element of another set.

Let the number of minutes of advertising be represented by x, and the cost of advertising be represented by y.

Given the function through which the cost of advertising can be determined. Therefore, the cost of advertising for the 10 min program will be when the value of x is 10. Therefore, the cost of advertising for 10 mins will be £100.

Hence, the cost of advertising for 10 minutes will be equal to £100.

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Which of the following equations results in infinitely many solutions?
9x80 = 8(x + 10)
9x + 10- 1x = 8(x + 10)
9x + 80 1x = 8(x + 10)
9x + 80 1x = 9(x + 10)

Answers

The equation with infinitely many solutions is:

9x + 80 - 1x = 8*(x + 10)

Which equation has infinitely many solutions?

An equation has infinitely many solutions when the dependence on x vanishes.

If you look at the third equation, we get:

9x + 80 - 1x = 8*(x + 10)

If we simplify the left side and expand the right side, we get:

8x + 80 = 8x + 80

So we have the exact same thing in both sides, this means that the equation is true for any value of x, so the equation has infinite solutions.

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The expression (x+5)(x+8)+(x+5)(2x-3) can be written as the product of (x+5) and

Answers

B=(x+5)
B(x+8)+B(2x-3)
B(3x+5)
(x+5)(3x+5)

To solve -3 = -19 + d/5, what steps would you use?

19 add divide multiply subtract

5 add divide multiply subtract

Answers

To solve the expression, the steps are add 19 and multiply by 5

How to solve the expression

Given the equation;

-3 = -19 + d/5

First,

Collect like terms

-3+ 19 = d/5

Add the values on the left side

We have

16 = d/5

Cross multiply

16 × 5 = d

We then have

d = 80

Thus, to solve the expression, the steps are add 19 and multiply by 5

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You enter a room. 2 dogs, 4 horses, 1 giraffe and a duck lie on the bed. 3 hens fly over a chair. How many legs are there on the floor?.

Answers

Answer:

4 legs from the bed and 4 from the chair im guessing since not all have 4

Hope This Help

Answer:

4 legs from the bed and 4 from the chair

Step-by-step explanation:

im a math wizard fr

Can someone help with these geometry fill in blank questions? Just 9, 10, 11, 12.. it’s urgent

Answers

I hope you can read this picture or answers below

9- AOD
10-AOD
11-BOD
12-BOC or angle 2

The graph of F(x), shown below, has the same shape as the graph of G(x) =
x², but it is shifted to the left 2 units. What is its equation?
F(x)=_
5
A. F(x)=x²+2
B. F(x)=x²-2
C. F(x) = (x-2)²
OD. F(x) = (x + 2)²

Answers

Answer: D

Step-by-step explanation:

See attached image.

What is the recursive formula for the geometric sequence with this explicit formula?

Answers

For the given geometric sequence, we have:

[tex]a_1 = 9\\\\a_n = (-1/3)*a_{n-1}[/tex]

So the correct option is B.

Which is the recursive formula?

Here we have the geometric sequence:

[tex]a_n = 9*(-\frac{1}{3} )^{(n - 1)}[/tex]

To get the initial value of the geometric sequence, we just need to replace n by 1, so we get:

[tex]a_1 = 9*(-\frac{1}{3} )*(1 - 1) = 9[/tex]

Now, notice that the common ratio is (-1/3), this means that each term of the sequence is (-1/3) times the previous term.

[tex]a_n = (-1/3)*a_{n-1}[/tex]

Then the correct option is B.

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I need help please..​

Answers

Answer:B

Explanation:In a function only one input is used there this is not a function

A rectangular container with a square base, an open top, and a volume of 1,372 cm3 is to be made. What is the minimum surface area for the container

Answers

The minimum surface area for the rectangular container is [tex]588cm^{2}[/tex].

How to find the surface area?

A solid object's surface area is a measurement of the overall space that the object's surface takes up.  Compared to the definition of the arc length of a one-dimensional curve or the definition of the surface area for polyhedra (i.e., objects with flat polygonal faces), where the surface area is equal to the sum of the areas of its faces, the mathematical definition of surface area in the presence of curved surfaces is much more complex.

Let s be the side of the square base and h be the height.

Surface Area=[tex]s^{2}+4sh[/tex]

Volume=[tex]s^{2}h[/tex]

According to the question,

[tex]s^{2}h=1372\\ h=\frac{1372}{s^{2} }[/tex]

So, surface area=[tex]s^{2} +4s(\frac{1372}{s^{2} })[/tex]

                          =[tex]s^{2}+\frac{5488}{s}[/tex]

Differentiate with respect to s,

Surface area=[tex]2s-\frac{5488}{s^{2} }[/tex]

Now, [tex]2s-\frac{5488}{s^{2} }=0[/tex]

[tex]2s=\frac{5488}{s^{2} } \\2s^{3}=5488\\ s^{3}=2744\\ s=14[/tex]

Find the value of h from the volume.

[tex]14*14*h=1372\\h=\frac{1372}{14*14}\\ h=7[/tex]

Thus, the minimum surface area=[tex]14^{2}+4*14*7[/tex]

                                                      =[tex]588cm^{2}[/tex]

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x^-4 for x=4

simplify this

Answers

Answer:

1/256

Step-by-step explanation:

4^-4 is the same thing as 1/4*1/4*1/4*1/4 which is 1/256

Answer:

1/256

Step-by-step explanation:

4⁻⁴ = 1/(4⁴) = 1/256

The storage container below is in the shape of a rectangular prism with a height of 6 feet and a length that is 2 feet more than its width.

Recall that the formula for the volume of a rectangular prism is V = l · w · h, where l is the length, w is the width, and h is the height.

Write the equation that represents the volume of the storage container in terms of its width.

Answers

The equation that represents the volume of the storage container(rectangular prism) in terms of its width is 12w + 6w²

How to find the volume of a rectangular prism?

volume of a rectangular prism = lwh

where

l = lengthw = widthh = height

Therefore,

h = 6 ft

l = 2 + w

Hence,

volume of the rectangular prism = (2 + w) × w × 6

volume of the rectangular prism = (2w + w²)6

volume of the rectangular prism = 12w + 6w²

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I kind of am," Fern said in a stage whisper while we were lining up waiting to jump in."Yeah, a bit," I admitted. I didn't want to tell her just how nervous. What if I panicked? What if I forgot how to breathe? Was that feeling ever going to go away? I would have loved to yank it out of myself and throw it into the sea.Actually, I had a lot of confidence in our training. On this shallow beginners' dive decompression wasn't going to be a concern. I knew my equipment would protect me if I got in trouble. And I'd be surrounded by helpful instructors. But even with all the fears on my mind, a few of my brain cells kept worrying about how I looked in my wetsuit. And that's how I knew I was going to be okay.Before I could get truly nervous though, it was Fern's turn to jump into the secret-keeping waves, then mine. We swam together to a buoy line and began pulling ourselves down the rope to the sandy sea floor. I noticed immediately that despite the dark water, I could actually see quite farand in color. Fern's mask was as vibrant as it had been on the boat. It did get a bit darker as we descended, but somehow it also got brighteras if the sea floor were reflecting what little light there was. When everyone was down, we all moved off toward the reef.Despite my nagging fear, I was mesmerized by this strange world. Everywhere my eye fell, there was something different to see. I'd never been surrounded by so much variety in shape, color, and texture. So many amazing creatures. Fish darted in and out of waving corals. Larger fish loomed, like harmless shadows. Small translucent shrimp-like creatures approached my mask. I could tell, by their quicksilver speed, that the littlest creatures were accustomed to the approach of larger predators.When Fern waved and pointed to a silvery green eel, sliding effortlessly toward an anemone while potential prey scattered before him, I realized that fear is natural. It helped all these creatures survive from day to day. The trick is learning to know when to use it to survive, and when it's only getting in the way of living. As I reached out to tickle a neon-colored sea slug, I certainly felt as if that trick were getting easier.Read this sentence from the passage:"I didn't want to tell her just how nervous. What if I panicked? What if I forgot how to breathe? Was that feeling ever going to go away? I would have loved to yank it out of myself and throw it into the sea."What would the narrator love to throw into the sea? the fish the excitement the nervousness the equipment Using the quadratic formula to solve 5x = 6x-3, what are the values of x? When the task of correctly identifying an individual in a slide of a four-person lineup was both difficult and important , participants in an experiment were especially likely to conform to others' wrong answers. This best illustrates the impact of: Pls help! Geometry Find the area