Verify the trig identity

The 4-th term in the sequence  d(n) = d(n - 1) × (-6) is -2592

Consider the mentioned recursive function of a sequence :

d(n) = d(n - 1) × (-6) and d(1) = 12

To find the fourth term of the sequence i.e., when n = 4,

Since the sequence is a recursive sequence, in order to find d(4) first we need to find d(2) and d(3)

For  n = 2,

⇒ d(2) = d(2 - 1) × (-6)

⇒ d(2) = d(1) × (-6)

⇒ d(2) = 12 × (-6)

⇒ d(2) = -72

For  n = 3,

⇒ d(3) = d(3 - 1) × (-6)

⇒ d(3) = d(2) × (-6)

⇒ d(3) = (-72) × (-6)

⇒ d(3) = 432

For  n = 4,

⇒ d(4) = d(4 - 1) × (-6)

⇒ d(4) = d(3) × (-6)

⇒ d(4) = 432 × (-6)

⇒ d(4) = -2592

Thus, the fourth term is -2592.

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The complete question is:

d(1) = 12,  

d(n) = d(n - 1) . (-6) ​

What is the 4-th term in the sequence?

The equation of the parabola with a vertex (2,3) and contains the point (0,0) is:  y = -3/4(x - 2)² + 3

Parabolic equitation or Quadratic equation with a vertex (2,3) and contains the point (0,0)  is written using the vertex form of the equation which is:

(x - h)² =  4P (y - k)

OR

standard vertex form, y = a(x - h)² + k     where a = 1/4p

The vertex

v (h, k) = (2, 3)

h = 2

k = 3

substitution of the values into the equation gives

y = a(x - 2)² + 3

considering the parabola passed the point (0, 0)

y = a(x - 2)² + 3

0 = a(0 - 2)² + 3

-3 = a(4)

a = -3/4

substituting for a

y = -3/4(x - 2)² + 3

hence the required equation is y = -3/4(x - 2)² + 3

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The expression is equivalent to 2, and option C is correct.

To simplify simply means to make anything easier. In mathematics, simplifying an equation, fraction, or problem means taking it and making it simpler. Calculations and problem-solving techniques simplify the issue. By eliminating all common factors from the numerator and denominator and putting the fraction in its simplest/lowest form, we can simplify fractions.

Given expression

[tex](\frac{4^{5/4} 4^{1/4} }{4^{1/2} } )^{1/2}[/tex]

using properties,

aⁿ/aˣ = aⁿ⁻ˣ

aⁿaˣ = aⁿ⁺ˣ

(aⁿ)ˣ = aⁿˣ

so expression is

[tex]({4^{5/4} 4^{1/4} }{4^{-1/2} } )^{1/2}[/tex]

adding powers

5/4 + 1/4 - 1/2 = 6/4 -1/2

5/4 + 1/4 - 1/2 = 1

substitute the values,

expression is, √4 = 2

Hence option C is correct.

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=====================================

Work Shown:

[tex](\text{x}-7)^2 = 36\\\\\sqrt{(\text{x}-7)^2} = \sqrt{36}\\\\\text{x}-7 = \pm\sqrt{36}\\\\\text{x}-7 = \pm6\\\\\text{x}-7 = 6 \ \text{ or } \ \text{x}-7 = -6\\\\\text{x} = 6+7 \ \text{ or } \ \text{x} = -6+7\\\\\text{x} = 13 \ \text{ or } \ \text{x} = 1\\\\[/tex]

-------------

Check:

Plug in x = 13

[tex](\text{x}-7)^2 = 36\\\\(13-7)^2 = 36\\\\(6)^2 = 36\\\\36 = 36 \ \ \checkmark\\\\[/tex]

This confirms x = 13

Now check x = 1

[tex](\text{x}-7)^2 = 36\\\\(1-7)^2 = 36\\\\(-6)^2 = 36\\\\36 = 36 \ \ \checkmark\\\\[/tex]

The value x = 1 is confirmed as well.

Both solutions are confirmed.

Amy will earn $26 in interest after 4 years.

Simple interest is a type of interest calculation where interest is only earned on the original principal (initial deposit) amount. It does not take into account any interest that has already been earned and is not compounded.

To calculate the simple interest earned on a deposit, you can use the formula: I = P * r * t, where I is the interest earned, P is the principal (initial deposit), r is the annual interest rate as a decimal, and t is the time in years.

In this case:

I = (100)(0.065)(4)

I = 26

The interest is: $26.

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Step-by-step explanation:

The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept.

Given that the slope of the line is -2/5 and the y-intercept is -2/5. We can substitute these values into the equation:

y = -2/5x + -2/5

y = -2/5x - 2/5

The equation in slope-intercept form is y = -2/5x - 2/5.

The smallest positive integer k such that 540k is a perfect cube is 8.

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Answer:

0.7901

Step-by-step explanation:

The equation has an imaginary root when (b) c = d

From the question, we have the following parameters that can be used in our computation:

2(c^2+d^2)x^2+(c-d)x+9=0

At the point of imaginary root, we have

b^2 < 4ac

This means that

(c - d)^2 < 4 * 2(c^2+d^2) * 9

So, we have

c^2 + d^2 - 2cd < 72c^2 + 72d^2

Evaluate the like terms

71c^2 + 71d^2 + 2cd > 0

Using a graphing calculator, we have

c = d

Hence, the solution is (b) c = d

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f(x) = -1/x is not a linear function.

Option (B) is correct,

What is a function?

A function is a mathematical relationship between two sets of quantities, called the domain and the range. In other words, a function is a rule that assigns to each element of a set (called the domain) exactly one element of another set (called the range). The most common way to represent a function is using an equation of form f(x) = y, where x is the input variable (or independent variable) and y is the output variable (or dependent variable).

A linear function is a function whose graph is a straight line. A linear function can be represented by an equation of the form f(x) = mx + b, where m and b are constants.

Out of the given options, the function which is not linear is B) f(x) = -1/x

This function is not linear because it's graph is not a straight line.

All the other options are linear functions, option A) f(x) = x/4 is a function with a slope of 1/4, option C) f(x) = 2x + 1 is a function with a slope of 2, and option D) f(x) = -3 is a constant function with a slope of 0.

Hence, f(x) = -1/x is not a linear function.

Option (B) is correct.

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The inequalities that include 7 in the solution set are;

D) -3 - x < -9

E) ¹/₂(x + 3) ≥ 2

A) -6 + x ≤ 5

Add 6 to both sides using addition property of equality to get;

x ≤ 11

B) 11 - x ≥ 7

Add x - 7 to both sides to get;

4 ≥ x

C) 3(2x - 9) > 15

Expand the bracket to get;

6x - 27 > 15

Add 27 to both sides to get;

6x > 42

Divide both sides by 6 to get;

x >  7

D) -3 - x < -9

Add 9 + x to both sides to get;

6 < x

E) ¹/₂(x + 3) ≥ 2

Multiply both sides by 2 to get;

x + 3 ≥ 4

Subtract 3 from both sides to get;

x ≥ 1

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The story: John requires 2 liters of oil for a trip. they have  3/4 liters of oil at home, how many liters of oil will he get from his father to make the trip?

x in the equation is the liters of oil required from the father. It was solved to be 1 1/4

In the problem, the equation made available is 3/4 + x = 2

The solution of the equation is as follows

3/4 + x = 2

collecting like terms

x = 2 - 3/4

x = 5/4

x = 1 1/4

The value of x = 1 1/4

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Answer: 5040

Step-by-step explanation: The equation for the volume of a pyramid is v=lhw/3

Answer:

Answer: 5040

Step-by-step explanation:

The volume of a triangular pyramid can be found using the formula V = 1/3AH where A = area of the triangle base, and H = height of the pyramid or the distance from the pyramid's base to the apex.

The completely factored form of 12x³-60x² + 4x - 20 can be expreesed as  4(3x² - 1)(x - 5).

The concept that will be used here is factorization. The process of breaking or decomposing an entity (like a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number or matrix is known as factorization or factoring.

To solve this problem then we need to take out all the common ratios :

12x³ - 60x² + 4x - 20

The we can factor 4 out as;

=[ 4(3x³ - 15x² + x - 5)]

The we can factorize the inner brackets  as:

[3x³ - 15x² + x - 5]

The we will have this

= [3x²(x - 5) + (x - 5)]

Then we can now express as

= (3x² + 1)(x - 5)

Therefore, option A is correct.

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Evan's earnings for the fuel he uses in the week are $1515.07597, which is more than $250. So Evan is correct that he will be paid more than $250 for the fuel he uses.

To convert miles to kilometers, we can use the conversion factor 1 mile = 1.609344 kilometers. Therefore, to convert the milometer readings to kilometers, we can multiply them by 1.609344:

Start of the week reading in kilometers: 120 miles * 1.609344 km/mile = 193.12128 km

End of the week reading in kilometers: 125465 miles * 1.609344 km/mile = 201788.50112 km

The distance Evan has traveled in the week can be calculated by subtracting the start of the week reading from the end of week reading:

Distance traveled in the week: 201788.50112 km - 193.12128 km = 7575.37984 km

Therefore, Evan has traveled 7575.37984 km in the week.

To calculate Evan's earnings for the fuel he uses, we can multiply the distance traveled in the week by the payment rate of 20 cents per kilometer:

Earnings for the week: 7575.37984 km * $0.20/km = $1515.07597

Therefore, Evan's earnings for the fuel he uses in the week are $1515.07597, which is more than $250. So Evan is correct that he will be paid more than $250 for the fuel he uses.

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The area of the rectangle is 20 square unit.

Let the length and width of the rectangle are respectively l and w.

Let another side of square is a.

Given that length of the rectangle is 5 more than the side of square.

l = a + 5

The width of the rectangle is 5 less than the side of the square.

w = a – 5

Given that the area of square is 45 square unit.

So, a² = 45

Now calculate the area of rectangle:

Area of rectangle = l * w

= (a + 5)(a - 5)

= a² - 25

= 45 – 25

= 20

Therefore, the area of rectangle is 20 square unit.

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Answer:

Step-by-step explanation:

Carla earned $27,500 during the year, while William attended law school full-time for 9 months and earned no income. They would not get dependent child care credit.

Credit for dependent care:

A married couple or single person who cares for his or her own or dependent child can earn this dependent care credit.

'The dependent tax credit is non-refundable.

* General credit qualifications.

- Employment-related care cost is required for a

= dependent under the age of 13 or

A dependent or spouse who lives with the taxpayer for more than half the year and is physically or mentally incapacited.

* amount of credit.

* applicable percentage - eligible care cost

- the appropriate percentage ranging from 20% to 35% depending on AGI

If your income is less than $15,000, the rate will be 35%. Income above $15,000 will decrease by 1% for every 2000.

* To get credit, married taxpayers must file a joint return.

*the cost of eligible care has been defined.

- costs for qualified individual care within the taxpayer's homr or outsider's home.

= if outside the home, the dependent or spouse must spend at least 8 hours per day within the taxpayer's home.

= unless the relative is a child under the age of 19, child care payments to a relative are eligible for the credit.

- The amount of costs that qualify is the lower of the actual cost or 3000 for one qualified individual and 6000 for two or more qualified individuals.

* Limitation on earned income.

- The amount of eligible care costs may not exceed the lower of the taxpayer's or spouse's earned income.

- Full-time students, disabled taxpayers, and spouses are deemed to have earned income up to the monthly limits of $250 for one child and $500 for two.

William and carla submit a combined tax return.

Carla earned $27,500 last year.

William attended law school full time for 9 months without earning any money.

They paid $3,500 for carl, their 3 year old child.

William and Carla are ineligible for dependent care credit. Because they should both have employment income.

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The value of x will be 6, while the value of y will be 3 in the given pair of the ordered equation.

As we have,

The pair of equations as:

2x + 8y = 36 -----------------(i)

5x − 4y = 18 -----------------(ii)

From above,

We can see that in the first equation the coefficient of x is 2 while in the second one, the coefficient of x is 5.

So, for solving the given sets of equation we need to first equalise the co-efficient of x in both the equations.

So, to make the coefficient equal we will multiply the first equation by 5 and the second equation by 2.

Thus we will get the set of equations as:

10x+40y=180--------(iii)

10x-8y=36------------(iv)

From (iii) and (iv), Now subtracting the (iv) from (iii),

We will get it as:

48y = 144

y = (144/48) = 3

Now for getting the value of x, we will put the value of y in the first equation.

Thus, we will get it as:

2x+24=36

=>2x=12

=>x = (12/2)=6

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Correct option is B, the given triangles are congruent by LL.

According to the leg-leg theorem (LL theorem), if the legs of two right triangles are the same length, then the complete triangle must be congruent.

Keep in mind that in this instance, the shorter (non-hypotenuse) sides of the triangle are referred to as the "legs." Right triangle sides all adhere to a particular pattern known as the Pythagorean theorem, which is why this theorem holds true.

According to the Pythagorean theorem, the length of the hypotenuse, squared, equals the length of each of the two legs, squared. Congruence states that two triangles must have equal hypotenuses and equal legs in order to be congruent.

Thus, the side-side-side (SSS) theorem, which was previously mentioned, is simply condensed into the leg-leg theorem.

In the given triangles,

the base of the triangles is same

the height of the triangles is common,

So, by Leg - Leg theorem, triangles are congruent.

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The explanatory and response variables for each set of data:

a.  number of hours spent reading is response variable and page number on which you are reading is explanatory variable.

b. calories burned is response variable and number of minutes of exercising is explanatory variable.

c. amount paid as income tax is response variable and the amount of a paycheck is explanatory variable.

d. pounds of hamburger used to make a meatloaf is response variable and number of people that can be fed from the meatloaf is explanatory variable.

A variable in mathematics is a symbol that represents a mathematical object. A variable can be a number, a vector, a matrix, a function, its argument, a set, or an element of a set.

Here,

For each set of data, the explanatory and response variables are as follows:

a. the number of hours spent reading is the response variable, and the page number on which you are reading is the explanatory variable.

b. The response variable is calories burned, and the explanatory variable is the number of minutes spent exercising.

c. the amount of income tax paid is the response variable, and the amount of a paycheck is the explanatory variable.

The response variable is the number of pounds of hamburger used to make a meatloaf, and the explanatory variable is the number of people who can be fed from the meatloaf.

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Cam can run 9.3205 miles when she finishes.

mile, any of several distance measurements, such as the statute mile of 5,280 feet (1.609 km). It was derived from the Roman mille passus, or "thousand paces," which was equivalent to 5,000 Roman feet.

The term "mile" is derived from the Latin "mille passus," which means "one thousand paces," and a mile was 1,000 Roman strides, with a stride equaling two steps. In 1592, the English Parliament standardised the Mile measurement at eight furlongs (660 feet).

Cam signed up to run a 15-kilometre running race.

So, here, we will convert kilometers to miles-

I kilometer is =0.621371 miles

And, I mile =1.609344 kilometres.

Thus, converting kilometres to miles.

Simply multiply the Number of kilometres by 0.62137.

Therefore, 1 kilometre =0.621371 miles

1.5 kilometer=15*0.621371 miles =9.32057 miles

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If the coin is tossed four times , then the probability of getting exactly two heads and two tails is 3/8   .

the  number of outcomes in tossing a coin is = 2 =  {H , T} ;

the number of outcomes in tossing the coin four times is = 2⁴ = 16 ;

we have to find the probability of getting exactly 2 heads and 2 tails :

So , the required outcomes will be = {HHTT, HTHT, HTTH, THHT, THTH , TTHH}  ;

the number of favorable outcomes is = 6 ;

the probability is = 6/16 = 3/8 .

Therefore , the required probability is 3/8 .

The given question is incomplete , the complete question is

If a coin is tossed 4 times, what is the probability of getting exactly two heads and two tails ?

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The population is all students in accounting class. 3.29 is sample statistic, not population parameter.

What is average ?

An average, also known as a mean, is a measure of central tendency that represents the typical value of a set of data. To calculate the average of a set of numbers, you add them up and then divide by the number of items in the set.

a. The statement "the population is all students enrolled in the accounting class" is true.

The population in this case refers to all the students who are currently enrolled in the accounting class and from which the sample is taken from.

b. The value 3.29 represents the sample statistic.

The sample statistic is a value that is calculated from a sample of data taken from the population. In this case, the average GPA of 3.29 is calculated using a sample of students enrolled in the accounting class, which is a subset of the population of all students enrolled in the accounting class. This value is used to make inferences about the population parameter, which is the true mean GPA of all students enrolled in the accounting class.

The population is all students in accounting class. 3.29 is sample statistic, not population parameter.

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Answer:

Simple interest is calculated by multiplying the principal (the initial amount of money deposited), the interest rate, and the number of years the money is on deposit.

To find out how much interest Diane will be paid in the first 2 years, we need to calculate the total interest earned over that time period.

First, we find the interest rate by multiplying the annual interest rate of 5% by .01 (to convert the percentage to decimal form)

0.05 * .01 = 0.05

Next, we find the total interest earned by multiplying the principal (400), the interest rate (0.05), and the number of years (2)

400 x 0.05 x 2 = 40

So, in the first 2 years, Diane will be paid $40 in interest on her deposit of $400.

Answer:

in total, Diane will be paid $20 + $40 = $60 in interest over the first 2 years.

Step-by-step explanation:

Simple interest is calculated as:

I = Prt

where:

I = Interest

P = Principal (initial amount deposited)

r = Interest rate (expressed as a decimal)

t = Time (in years)

So in this case, with a principal of $400 and an interest rate of 5% (or 0.05 as a decimal), the interest paid in the first year would be:

I = $400 * 0.05 * 1 = $20

And in the second year, it would be:

I = $400 * 0.05 * 2 = $40

So in total, Diane will be paid $20 + $40 = $60 in interest over the first 2 years.

A line that is perpendicular to the line y = -1/3x + 7 has a slope that is the negative reciprocal of -1/3 which is 3.

We know that the line intersects the point (4,2), so we can use this point and the slope to write the equation of the line in point-slope form:

y - y1 = m(x - x1)

where (x1, y1) is the point the line passes through, m is the slope, and y and x are the coordinates of any point on the line.

So the equation of the line that is perpendicular to y = −1/3x + 7 and intersects the point (4,2) is:

y - 2 = 3(x - 4)

Simplifying this, we get

y = 3x - 2

To convert this to the slope-intercept form we can rewrite it as

y = 3x + b

Therefore, the equation of the line that is perpendicular to y = −1/3x + 7 and intersects the point (4,2) is y = 3x - 2

The table using this function rule is:

 x      y

-1     -9

 0     1

 1      11

 5     51    

When the specified input values are applied, the function rule y = 2x+4 produces the output values displayed in the table. As of right now, the output numbers are denoted by and the input values are denoted by and. The function rule y = 3 x + 1 can therefore alternatively be represented as f (x) = 3 x + 1.

The connection between the input or domain and the output or range is called a function rule. When there is exactly one value in the range for each domain value, a relation is a function.

y = 10x+1

x = -1→y=10(-1)+1=-10+1→y=-9

x = 0→y=10(0)+1=0+1→y=1

x = 1→y=10(1)+1=10+1→y=11

x = 5→y=10(5)+1=50+1→y=51

x      y

-1     -9

0     1

1      11

5     51    

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The complete question is,

Use this function rule to complete the table's fields.

y = 10x + 1.