For, n = 0, 1 we proved that 7⁽ⁿ⁺⁴⁾ - 7ⁿ hence by induction it is divisible by 30 for all n ∈ N.
What is mathematical induction?An inductive proof requires two cases. The first, or base case, establishes the proposition that n = 0 without requiring any prior knowledge of other situations. The second example, the induction step, demonstrates that if the assertion is true for any particular case where n = k, then it must also be true for the subsequent case when n = k + 1. These two actions prove that the statement is true for all n = 1 natural numbers. The base case establishes the truth of the statement for all natural numbers n N, but it does not always start with n = 0. Instead, it frequently starts with n = 1, and it may also start with any fixed natural number n = N.
Given, We have to prove 7⁽ⁿ⁺⁴⁾ - 7ⁿ is divisible by 30 for all n ∈ N.
Let, n = 0.
Therefore,
7⁴ - 7⁰.
= 2401 - 1.
= 2400.
= 30×80, Divisible by 30.
Similarly for n = 1 we have,
7⁵ - 7¹.
= 16807 - 7.
= 16800.
= 30×560.
Hence by induction, the cycle will continue.
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-15 POINTS-
How many sides does a regular polygon have if one interior angle measures 144°
Answer: 10 sides
Step-by-step explanation:
You can always use the given formula to calculate but I (personally) find it much easier to calculate the number of sides using the exterior angle.
To calculate the exterior angle, it is 180 - the interior angle.
In this example, the exterior angle would be 180-144 = 36°
Because it is a regular polygon, we know that all the angles in the shape are the same.
In all polygons, the exterior angles always add up to 360 degrees.
So divide 360 by 36 (the size of each exterior angle), and you get 10.
360/36 = 10
Find three consecutive integers such that 4 times the first decreased by the second is 12 more than twice the third
The three consecutive number are 17 , 18 and 19 respectively
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Let the first number be = x
Let the second number be = ( x + 1 )
Let the third number be = ( x + 2 )
Now ,
A = 4 times the first decreased by the second is 12 more than twice the third
Substituting the values in the equation , we get
4x - ( x + 1 ) = 12 + 2 ( x + 2 )
On simplifying the equation , we get
4x - x - 1 = 12 + 2x + 4
3x - 1 = 16 + 2x
Subtracting 2x on both sides of the equation , we get
x - 1 = 16
Adding 1 on both sides of the equation , we get
x = 17
Therefore , the value of x is 17
Hence , the consecutive numbers are 17 , 18 and 19
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Help me pleaseeeeeee
please answer this is due at 11:59PM!!!
Answer:
Step-by-step explanation:
An explanatory variable is what you manipulate or observe changes in while a response variable is what changes as a result (e.g., reaction times). The words “explanatory variable” and “response variable” are often interchangeable with other terms used in research.
-50 POINTS- don’t explain
Answer:
3
Step-by-step explanation:
like you said don't explain
WILL MARK BRAINLIEST given angles A and B in quadrant I with tan = 7/24 and tan B = 15/8, find tan (A + B), find the exact value of each expression by applying the identity.
Using PEMDAS to resolve the trigonometric identity which involves trigonometric function, the value of tan A + tan B is 416 / 87
What is Trigonometric IdentityA trigonometric identity is an equation that is true for any value of the variables involved. These identities are useful in solving problems involving trigonometric functions. Examples include the Pythagorean identity, the sine and cosine addition formulas, and the double angle formulas.
In the given problem, we have a compound angle problem and we can solve it as;
tan A + tan B = [(tan A + tan B)] / [1 - tan A * tan B]
tan A = 7/24tan B = 15 / 8Substituting the values into the expression and solve;
tan A + tan B = [(7/24 + 15/8)] /[1 - (7/24)(15/8)]
Applying the rules of PEMDAS;
tan A + tan B = (13 / 6) / (29 / 64)
tan A + tan B = 416/87
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A new pair of shoes you would like to purchase is 65% off. If you paid $135 for the shoes, how much was the original cost?
Solving the equation the original cost of the shoes before the discount will be obtained as $385.
To find the original cost of the shoes before the discount, you can use the following equation:
original cost = final cost÷ (1 - discount percentage)
In this case:
original cost = $135 ÷ (1 - 0.65)
To find the final cost you need to multiply 135 by 1 - 0.65.
0.65 is the equivalent to 65% when it's represented as a decimal.
0.65 represents the proportion of the original price that was removed (65% or 0.65)
So:
original cost = $135 ÷ (1 - 0.65)
original cost = $135 ÷ (1 - 0.65) = $135 ÷ 0.35 = $385
The original cost of the shoes before the discount was $385.
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What's the inverse of ƒ(x) = x – 20?
Therefore , the solution of Inverse function comes out to be of x-20 equals to x+20
What is the inverse symbol?A function f has an inverse function only if for every y in its range there is only one value of x in its domain for which f(x)=y. This inverse function is unique and is frequently denoted by f−1 and called “f inverse.” For an overview into the idea of an inverse function, see the function machine inverse.Steps for finding the inverse of a function fReplace f(x) by y in the equation describing the function. Interchange x and y. In other words, replace every x by a y and vice versa.
Here,
Solve for y.
Replace y by f-1(x).
y=x-20
x=y-20
Solve for y,x = y-20
x+20
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Answer:
ƒ^(-1)(x) = x + 20
Step-by-step explanation:
To find the inverse of the function ƒ(x) = x - 20, we need to switch the roles of x and y and solve for y.
Let's start by replacing ƒ(x) with y:
y = x - 20
Next, swap x and y:
x = y - 20
Now, solve for y by isolating it:
x + 20 = y
Therefore, the inverse of the function ƒ(x) = x - 20 is given by:
ƒ^(-1)(x) = x + 20
Given that c:d is 4:3 and d:u is 2:9, find c:d:u.
Answer:
c : d : u = 8 : 6 : 27
Step-by-step explanation:
Given c:d = 4:3 and d:u = 2:9, you want c:d:u.
Ratio unitsTo write the desired ratio, we need to convert the others so they have the same number of ratio units representing 'd'. That number can be the least common multiple of 3 and 2, which is 6. Knowing this, we can write ...
c:d = 4:3 = 8:6
d:u = 2:9 = 6:27
Then the desired ratio is ...
c:d:u = 8:6:27
a restaurant offers 7 appetizers and 9 main courses. In how many ways can a person order a two-course meal
3) At the grocery store, the bags of oranges weigh 4 1/3 pounds. How much would 2 1/2 bags of oranges weigh?
Answer:7 7/8. pounds.
Step-by-step explanation:
What is the solution of the inequality shown
below?
a+5>-7
The required solution for the inequality a + 5 > -7 is (-12, ∞).
What is inequality?Inequality shows relation between two expression which are not equal to each others.
The given inequality is,
a + 5 > -7
To find the solution for the inequality,
Simplify the inequality,
a + 5 > - 7
a > - 7 - 5
a > -12
The value of a is greater than -12, implies that,
The solution for the value of the given inequality is (-12, ∞).
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Line segment AB = 12, its image after a dilation is 36. What is the scale factor of the dilation?
The scale factor of the dilation is 3
How to determine the scale factor of the dilation?From the question, we have the following parameters that can be used in our computation:
Line segment AB = 12 units
After the dilation = 36 units
The scale factor of the dilation is then calculated as
Scale factor of the dilation = After the dilation/Line segment AB
Substitute the known values in the above equation, so, we have the following representation
Scale factor of the dilation = 36/12
Evaluate
Scale factor of the dilation = 3
Hence, the scale factor is 3
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is 0.2/2.6 0.3/3.6 proportional or not proportional
Answer:
not proportional
Step-by-step explanation:
0.2/2.6 is 1:13
0.3/3.6 is 1:12
(1/a)^3 Help simplify please
Answer:
1/a³
Step-by-step explanation:
(1/a)³ = 1³/a³ = 1/a³
1. The height of a right triangular prism is 1 5/6 inches. Each side of the triangular base measures 10 inches, and the height of the base is 8 2/3 inches. The triangular prism is placed atop a cube whose side measures 10 inches so that one of the triangular prism’s bases lies completely on one side of the cube.What is the surface area of the solid formed?
100 POINTS!!! IF YOU DRAW AN ACCURATE DIAGRAM TOO, YOU WILL BE AUTOMTIC BRAINLIEST!!!!!!
If the height of a right triangular prism is 1⁵/₆ inches. The surface area of the solid formed is 598.33 in².
What is surface area?The surface area of a solid object is a measure of the total area that the surface of the object occupies.
A three-dimensional solid form with six faces, including rectangular bases, is called a rectangular prism. A rectangular prism also refers to a cuboid. A cuboid and a rectangular prism have the same cross-section.
First step:-
Right triangular prism=1⁵/₆ inches= 11/6 inches
Height of the base=8²/₃ inches = 26/3 inches
Second step:-
Surface area = 5(10× 10) + (0.5÷ 10 × 26/3) + 3(10× 11/6)
Surface area = 5(100) +43.33+ 3(18.33)
Surface area = 500) +43.33 + 55
Surface area = 598.33 in²
Therefore the surface area of the solid formed is about 598.33 in²
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The gray figure below is the image of the black figure after a dilation. Which represents the dilation?
A (x,y) -> (2x, 2y)
B (x,y) -> (3x, 3y)
C (x,y) -> (0.5x, 0.5y)
D (x,y) -> (x+4, y+5)
The answer is A (x,y) -> (2x, 2y) it is representing the dilation.
What is dilation?Dilation is a medical procedure that involves widening a part of the body, such as the cervix or esophagus, by stretching and enlarging it. This can be done for a variety of reasons, including to allow for the passage of a baby during childbirth or to facilitate the insertion of a medical instrument or device. Dilation is usually done using dilators, which are cylindrical tools with a tapered end that can be inserted into the body and then gradually increased in size.
Dilation is a geometric transformation that enlarges or reduces the size of a shape by a factor. In this case, the shape is enlarged, with the resulting shape being twice the size of the original black figure. This is represented by the equation A (x,y) -> (2x, 2y), where the x and y coordinates of each point in the figure are multiplied by two.
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hat is the following sum?
0 7(10√/x²y)
O
O
0 7(²√x²¹x²)
The sum of the given equation is 7[tex]\sqrt[5]{x^{2} y}[/tex].
What is fractional equation?A fractional equation is one in which one or more of its terms have the unknown as their denominator. Equations involving fractions can be resolved using the Cross-Product property.Therefore, when we reduce or simplify a fraction, we aim for maximum simplicity. In order to do this, we divide both the numerator and the denominator by the biggest integer that can be divided precisely into both numbers. In other words, we choose the greatest number that the top and bottom have in common and divide them by that.Equations using fractions require the solution of equations in which the unknown variable is a component of the fraction's numerator or denominator. To solve fractional equations, we must determine the value of the undetermined variable.Given data :
4 ([tex]\sqrt[5]{x^{2} y}[/tex].) + 3 ([tex]\sqrt[5]{x^{2} y}[/tex].)
Add similar elements
= 7[tex]\sqrt[5]{x^{2} y}[/tex].
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Peter is building a ramp at his home. He wants to build it 11 inches high with aa 60 inch base. How long will the ramp be that Peter is building? Round to the 10ths if needed.
Explanation:
You have a right triangle with legs of 11 and 60. The hypotenuse is unknown.
Use the pythagorean theorem to find the hypotenuse.
[tex]a^2 + b^2 = c^2\\\\11^2 + 60^2 = c^2\\\\121 + 3600 = c^2\\\\3721 = c^2\\\\c^2 = 3721\\\\c = \sqrt{3721}\\\\c = 61\\\\[/tex]
The ramp is exactly 61 inches long.
How do you find the equation of the midline for a function?
Therefore , the solution to the given problem is the equation of the midline is: y = M+m /2 .
What is equation?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equality of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to express the connection between two phrases on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.
Here,
The midline equation is y = M+m /2
when a function has: o a max M AND a min m or o (if it does not have extrema) a bound above M and a bound below m, where M is the minimum bound above and m is the maximum bound below.
When one of the following conditions is true for a function, even though it has a bound below, there is no midline (even if it has a bound above)
It is solely constrained from above (or below)
Any line is midline if a function has the values (-∞, +∞)
Therefore , the solution to the given problem is the equation of the midline is: y = M+m /2 .
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Explain what the ratio 15:73 represents
Therefore , the solution of the given problem of ratio comes out to be 15:73 ratio simplified is 15/73.
What is ratio?A ratio or fraction in math displays how many times one number is contained in another. For instance, the proportion of oranges to lemons in a bowl of fruit is eight to six if there are eight oranges and six lemons (that is, 8:6, which is equivalent to the ratio 4:3 or decimal 1.333).
Here,
To simplify the ratio 15:73, we find the greatest common divisor of 15 and 73, and then we divide 15 and 73 by the greatest common divisor.
The greatest common divisor that you can use to simplify 15:73 is 1. Which means the answer to ratio 15:73 simplified is:
Here is the math to illustrate better:
15/1 = 15 and 73/1 = 73
Thus, 15:73 ratio simplified is 15/73
Therefore , the solution of the given problem of ratio comes out to be 15:73 ratio simplified is 15/73.
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Jake has a floor that is 64 square feet. He wants to tile the entire floor using large tiles that measure 3 feet by 5 feet. The tiles may be cut as needed. What is the fewest number of tiles required to fill the space while also requiring the fewest tiles to be cut? Show a diagram of the completed floor.
The number of tiles required to fill the space while also requiring the fewest tiles to be cut is 5 .
What is area ?
Area can be defined as the total space taken by a 2-dimensional surface or shape of an object.
Given
Jake has a floor that is 64 square feet.
He wants to tile the entire floor using large tiles that measure 3 feet by 5 feet.
The area of 3 feet by 5 feet could be = 3 * 5 = 15 square meters.
number of tiles required to fill the space could be = 64 / 15
that is 4.3 (approx )
Hence , the number of tiles required to fill the space while also requiring the fewest tiles to be cut is 5 .
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XYZ Company
Balance Sheet
Assets
Cash
$45,000
Inventory $41,000
Property $134,000
Total Assets:
Liabilities
Notes
Wages
Equity
Stock
$55,000
$56,000
$50,000
Investment $59,000
Total Liabilities + Equity:
Using the balance
sheet, calculate XYZ
Company's total
assets - liabilities.
Total Assets - Liabilities = $ [?]
Enter
PH
Skip
On solving the provided question, we can say that - by formula of simple interest we have SI = [tex]2200 X 2 X 4/100 = > 17600/100 = > 176[/tex]
what is interest ?Multiplying the principal by the interest rate, time, and other factors yields simple interest. Simple return equals principle times interest times hours is the marketed formula. It is easiest to compute interest using this formula. A percentage of the principle balance is how interest is most commonly computed. For instance, he will only pay his portion of the 100% interest if he borrows $100 from a buddy and promises to pay it back with 5% interest. $100 (0.05) = $5. When you borrow money, you must pay interest, and when you lend money, you must charge interest. Most of the time, interest is expressed as a yearly percentage of the loan amount. The interest rate on the loan is known as this percentage.
here,
principal, p = 2200
rate, r = 2%
time, t = 4 rs,
so SI = P X R X T/100
SI -
[tex]2200 X 2 X 4/100\\17600/100\\176[/tex]
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A square board is attached to a wall. The board is divided into four squares and each square is painted either black or white. In how many ways can the board be painted if we can rotate the given board about its center? The answer is not 14 or 16.
The board can be rotated in 4 different ways (0, 90, 180, 270 degrees) and each of the four squares can be painted in 2 different ways (black or white). Since we can rotate the board, the order in which the squares are painted does not matter, so we must divide by the number of ways to rearrange the squares. So, there are (2^4) / 4 = 6 ways the board can be painted when we take rotations into account.
How to calculate the Rotation rate?To calculate turnover, it is necessary to consider the departures and additions of people to the company in the period. Calculating turnover is quite simple: you need to add the number of admissions and the number of dismissals, then just divide by 2 and then divide by the total number of employees within your company.
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Beth poured 1/2 of the juice from a 5-liter bottle while serving guests at a party. How much juice, in liters, is still left in the bottle?
Answer:
2.5 litres
Step-by-step explanation:
a recipe says 4 cups of flour makes up to 280 mini cupcakes. If you want to make 120 mini cupcakes how many cups of flower do you need?
What is the unit rate for 4543.08 km in 52.4 h?
The unit rate for 4543.08 km in 52.4 hours is 86.7 km.
What is a unit rate?It is the quantity of an amount of something at a rate of one of another quantity.
In 2 hours, a man can walk for 6 miles
In 1 hour, a man will walk for 3 miles.
We have,
4543.08 km = 52.4 hours
Divide 52.4 into both sides.
86.7 km = 1 hour
Thus,
The unit rate is 86.7 km.
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State the domain and range of the following function.
Answer:
Step-by-step explanation:
ur STUP1d
\int (2)/(\sqrt(5x^(2))+3)dx and steps
Answer:
Step-by-step explanation:
1. A student has computed that it takes an average of 17 minutes with a standard deviation of 3 minutes to drive from home, park the car, and walk to an early morning class. Find the proportion of times whose time X satisfies each of the following conditions:
A) If you selected 10 days at random, what is the probability that half the days will take between 15 and
20 minutes?
B) What is the probability the second day selected will be the first day he is late to class, assuming he left
at 8:39am?
(I know how to solve for the probabilities on their own, however I’m not sure how to incorporate the added probability based on multiple days)
Answer:
B
Step-by-step explanation: