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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To solve -3 = -19 + d/5, what steps would you use?
19 add divide multiply subtract
5 add divide multiply subtract
To solve the expression, the steps are add 19 and multiply by 5
How to solve the expressionGiven 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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4/15 divided by 1 1/3 + 0.4
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
Can someone help with these geometry fill in blank questions? Just 9, 10, 11, 12.. it’s urgent
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)
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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What is the recursive formula for the geometric sequence with this explicit formula?
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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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.
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 = heightTherefore,
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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Please help ill give whoever answers the best the brainliest answer :)))
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.
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?.
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
Which phrase best describes the translation from the graph y = (x – 5)2 + 7 to the graph of y = (x + 1)2 – 2?
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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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
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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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.
x^-4 for x=4
simplify this
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
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.!
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 statistic is used to determine if there is a significant difference between three or more sets of related scores
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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Use the table to find the products of the two polynomials. Write your answer in
descending order. B) (x2 + x – 2)(4x2 – 8x)
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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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?
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.
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
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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The expression (x+5)(x+8)+(x+5)(2x-3) can be written as the product of (x+5) and
Given the following information, determine which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
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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Which of the following is a fifth root of the given complex number?
Answer:
D
Step-by-step explanation:
This result is true by De Moivre's theorem.
help aaahhhh
x =x=x, equals
^\circ
∘
degrees
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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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
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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At what values of x, does f(x) = 0?
a. 0
b. -1
c. -8
d. 4.2
e. 4
f. 2
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
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
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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HELP ASAP NO ROCKY!!!!!!!!
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.
Please help! Which is the solution set of the following inequality?
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
I need help please..
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
Answer: D
Step-by-step explanation:
See attached image.
someone please please help me with this !!
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).
Subtract 6x^{2}-7x-2 from 6x^{2}-2x+6
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!