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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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The number of tiles Lynn needs to cover the floor with an area of 16.875 ft² is 17 tiles

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

The floor measures 4 1/2 ft by 3 3/4 ft.

Hence:

Area of floor = 4 1/2 ft * 3 3/4 ft = 9/2 ft * 15/4 ft = 16.875 ft²

Each tile measures 1 square foot each, hence:

number of tiles needed = 16.875 ft² / 1 ft² per tile = 16.875 tiles ≅ 17 tiles

The number of tiles Lynn needs to cover the floor with an area of 16.875 ft² is 17 tiles

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Answer:

Step-by-step explanation:

First, since they have the same base, you can join them. Because it is division, you have to subtract the exponents.

4^(-11/3 -(-2/3)) = 4^-9/3 = 4^-3

Negative exponents turn the base into a fraction and 4^3= 64

so the answer is C. 1/64

Based on the cost of each of the snacks in a s'more, the number that the summer camp counselor can make is E.10.

The total change in temperature on the surface of Mercury is C. 1,168°F.

First, find out the cost of one s'more:
= (2 x 0.50) + 0.75 + (2 x 1.25)

= 1 + 0.75 + 2.50

= $4.25

The total s'mores that can be made is:

= 45 / 4.25

= 10 complete s'mores

= 869 - (-299)

= 869 + 299

= 1,168°F

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Answer:

6 servings

Step-by-step explanation:

Box = 72 cups

1 serving size = 12 cups

72 ÷ 12 = 6

The length of the Rectangle is 9x +6 , Option D is the right answer.

A rectangle is a polygon with four sides , four angles and four vertices , The opposite sides are parallel and equal .

An expression representing the area of Rectangle is given

One of the side is 5x -8

The other side =  ?

As Area of Rectangle =  Length * Breadth

It can be calculated by Dividing Area by Breadth

(45x² - 45x -48 ) / (5x -8)

= 9x +6

Therefore the length of the Rectangle is 9x +6 ,Option D is the right answer.

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Answer:

9x + 6
INCLUDED WHOLE PAGE!!
☆
EDGE2022; Good Luck :3!!!

The inequality for sec (x) < cot (x) is;  π/2 < x < π

We are given that;

We want to find the intervals that the trigonometric inequality sec (x) < cot (x) always hold true.

This can also be expressed as;

1/cos (x) < 1/tan (x)

Now, this can happen only in the quadrant where tan (x) is negative and cos x is positive which is in fourth quadrant where;

π/2 < x < π

The inequality for sec (x) < cot (x) is;  π/2 < x < π

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Answer:

C on edge 22 for Pre-Calculus

Step-by-step explanation:

pi < x< 3pi/2

I retook the unit test, put C, and got a 100. I promise it's right

The total number of digits used when the  first 2002 positive even integers are written is 4 digits

Using arithmetic sequence,

nth term = a + (n - 1)d

where

a = 2

d = 2

nth term = 2 + (2002 - 1)2

nth term = 2 + 4002

nth term = 4004

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By the ASA postulate, it can be said that the triangle ABC and triangle MRQ are congruent. Thus, option B is correct, Yes, they are congruent by either ASA or AAS.

The angle sum property of a triangle states that the sum of the interior angles of a triangle is 180 degrees.

Using Angle Sum Property, in ΔMRQ

<M + <R + <Q= 180

<M + 42 + 85 = 180

<M = 180 -42

<M= 53°

Now, In ΔMRQ and ΔABC  

<A= <R ,

<B = <M

AB = MR

Hence, By ASA criteria ΔMRQ ≅ ΔABC.

Thus, option B is correct, Yes, they are congruent by either ASA or AAS.

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Answer: Yes, they are congruent by either ASA or AAS.

Step-by-step explanation:

Answer:

[tex]13.86[/tex]

Step-by-step explanation:

you should see 2 triangles with angle 60 bounded by two equal sides of 4

[tex]=2(\frac{1}{2} (4)(4)sin 60)\\=13.86[/tex]

The linear model is 12 + x = 0.80(20 + x)

Aditi would need to win is 20.

Given this linear model: 12 + x = 0.80(20 + x)

Take the following steps in order to determine the value of x:

Expand the bracket : 12 + x = 16 + 0.8x

Combine like terms: 16 - 12 = x - 0.8x

Add similar terms: 4 = 0.2x

Divide both sides by 0.2 : 4/ 0.2

x = 20

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Answer:

the first answer would be C and the second is B I  think

Step-by-step explanation:

based off what I just did so 20 is the second answer aka B lol

Answer:

6xp^24

Step-by-step explanation:

Basically, there is 24 6xp's which is 6xp^24.

Hope I explained well enough.

The range of the function on the graph is (c) all real numbers less than or equal to 3

The missing graph is added as an attachment

From the graph, we have the following highlights

This means that the range is (c) all real numbers less than or equal to 3

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Answer:

V = 262.5 ft.

Step-by-step explanation:

Volume = length × width × height

6.25 × 14 × 3

6.25 × 42

262.5

The condition that must not be met before usage us that the coefficient of the x²-term must be positive.

The standard quadratic function is given as;

ax²+bx + c = 0

From the equation shown above, the leading degree of the equation is 2. The following are condition that must be met before you can use

the quadratic formula to find the solutions of an equation.

The condition that must not be met before usage us that the coefficient of the x²-term must be positive.

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Answer:

Q1: 62

Q2: 13

Step-by-step explanation:

Just follow Order of operations, and you get the answers.

P arenthesis

E xponents

M ultiplication

D ivision

A ddition

S ubtraction

(B) The median for the students is 2 and the median for the teachers is 3, comparing the medians of the data.

Therefore, the median for the students is 2 and the median for the teachers is 3, comparing the medians of the data.

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Correct question:

Which compares the medians of the data?

(A) The median for the students is 4 and the median for the teachers is 8.

(B) The median for the students is 2 and the median for the teachers is 3.

(C) The median for the students is 4 and the median for the teachers is 3.

(D) The median for the students is 2 and the median for the teachers is 8.

The inequality that represents the range of k so that inequality 1 has real solution is -3 < k < 5

Inequality 1

x^2 + (k + 1)x + k + 4 > 0

The discriminant is calculated using

D = b^2 - 4ac

So, we have:

D = (k + 1)^2 - 4 * 1 * (k + 4)

Evaluate

D = k^2 + 2k + 1 - 4k - 16

This gives

D = k^2- 2k - 15

Factorize the equation

D = (k+ 3)(k -5)

Solve for k

k = -3 and k = 5

Express the k values as inequalities

-3 < k < 5

Inequality 2

The inequality cannot be read, because of the image quality

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there are total 64 is the answer if she puts 2 cookies in each packet

hope this helps u :)

Answer:

1578 + 1000 = 2578

2578 + 1000 = 3578

3578 + 1000 = 4578

4578 + 1000 = 5578

5578 + 1000 = 6578

Answer: [tex]7\pi[/tex]

Step-by-step explanation:

The area of the entire circle is [tex](\pi)(3^{2})=9\pi[/tex]

The area of each of the smaller circles is [tex](1^{2})(\pi)=\pi[/tex], so their combined area is [tex]2\pi[/tex].

By subtracting areas, we get the area of the shaded region is [tex]7\pi[/tex].

jill is 24 years old.

The ratio of vina's age to jill is 7:6 .

      Let vina's age =7x

      let jill's age = 6x

If vina is 4 years older than jill.

Then ,                   7x - 6x = 4

                                  x = 4

Now  vina's age = 7x = 7 * 4 = 28 years

          jill's age = 6x = 6 * 4 = 24 years.

Therefore, jill is 24years old.

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The probability of a royal flush in a 5-card hand is 1/649740 or 0.0001539%.

The combination is a way of selecting a smaller number of sets from a larger number of sets in no particular order.

If selecting r items from a set of n items, in no particular order, we use combinations, using the formula:

nCr = n!/{r!(n - r)!}.

In the question, we are asked the probability of a royal flush in a 5 card hand.

The number of ways of selecting 5 cards from a deck can be computed using a combination, 52C5, as we have 52 cards in a deck, 5 cards are to be chosen and the order is not in concern.

52C5 = 52!/{5!(52 - 5)!} = 52!/(5!.47!) = (52*51*50*49*48)/(5*4*3*2*1) = 2598960.

The number of hands that make a royal flush is only 4. 4 from each suit.

Therefore, the probability = 4/2598960 = 1/649740 or 0.0001539%.

Therefore, the probability of a royal flush in a 5-card hand is 1/649740 or 0.0001539%.

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Answer:

-1 , 4

Step-by-step explanation:

x = ( -2 + 0 ) ÷ 2

y = ( -1 + 9 ) ÷ 2

-2 ÷ 2 = -1

8 ÷ 2 = 4

( -1 , 4 )

The day of the month that was 75% of the pond algae-free is: on day 28 or third day.

Since we were told the algae  doubled every day until it was  covered in algae on day 30 of the month.

Hence:

Day 30 (First day)= It was 100% covered.

Day 29 (Second day)=It was 50% covered.

Day 28 (Third day )=it was 25% covered, which will be the same as 75% pond algae-free.

Therefore the day of the month that was 75% of the pond algae-free is: on day 28 or third day.

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The complete question is:

amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day 30 of the month. On what day of that month was 75% of the pond algae-free?

62.5 seconds

Step-by-step explanation:

The height of the tower is 452.66 ft

The angle formed by the line of sight and the horizontal plane for an object above the horizontal.

The angle of depression is the angle between the horizontal line and the observation of the object from the horizontal line.

Given:

we have two angles one of which is angle of elevation and another angle depression.

Using trigonometric ratio,

tan ∅ = P / B

tan 32° = h / 425

0.624= h /425

h= 265.569 ft

again,

tan 29° = h(bottom of the tower) / 425

h(bottom of the tower) = 425 × tan 29°

h(bottom of the tower)  = 235.581 ft

Thus, Height of tower = h(top of the tower) + h(bottom of the tower)

Height of tower = 265.569  + 235.581

Height of tower = 501.15 ft

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