What is the sum of the geometric series?

Sigma-Summation Underscript n = 1 Overscript 4 EndScripts (negative 2) (negative 3) Superscript n minus 1
–122
–2
40
54

180 - 115 = 65

4k + 5 + 6k + 10 + 65 = 180

10k + 80 = 180

10k = 100

k = 10

When using percent slope to get tree height, the tree is the rise, and the horizontal distance from the tree with the ground is the run. We can simply measure our horizontal distance from the tree, and we have instruments for gauging the percent slope to the top of a tree.

Hence, with those two measures (run and %slope) we can solve for rise.

You Total up all the trees.

let the average of number of trees are:

3+3+8+10+5+2=31

then you subtract the amount of trees under 75 feet in height.

31-8-3-3=14

divide the number of trees 14 under 75 ft in height by the total amount of trees31.

14/31=0.4516

0.4516=45.2

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for any value of n, the nth term of the equation will be:

Aₙ = -3* 2ⁿ⁻¹

A geometric sequence is a sequence of numbers that follows a pattern where the next term is found by multiplying by a constant called the common ratio, r.

From the general formula of the nth term of the geometric sequence:

Aₙ = A₁* rⁿ-1

Given a geometric sequence:

-3, -6, -12, 24,.....

From the general formula of the nth term of the geometric sequence:

Aₙ = A₁* rⁿ⁻¹

in our case

a₁ = -3

ratio(r) = -6/-3 = 2

Thus for any value of n, the nth term of the equation will be

Aₙ = -3* 2ⁿ⁻¹

For instance,

8th term of the given sequence:

A₈ = -3* 2⁸⁻¹

A₈ = -3* 2⁷

A₈ = -384

Therefore, for any value of n, the nth term of the equation will be

Aₙ = -3* 2ⁿ⁻¹

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George is not correct because we can find the least common multiple of 8 and 6 which is 24. This allows us to solve the problem with simple numbers and ease our calculations to get the answer in the simplest form.

The equation solved by George is 3/8 - 1/6 = 18/48 - 8/48

To solve the given fractions in the simplest form, we will find the common denominator which would be the least common multiple of both denominators.

we can see that the numbers in the denominator are 8 and 6.

Multiples of 8: 8, 16, 24, 32, 40,...

Multiples of 6: 6, 12, 18, 24, 30,...

Hence, LCM of 8 and 6 is 24.

thus we can solve the equation as follows -

3/8 - 1/6

= 3*3/8*3 - 1*4/6*4

= 9/24 - 4/24

= 5/24

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The slope of a line in the xy plane b = 13/3

The slope of a line in the xy plane can be calculated by using the equation m = (y2-y1)/(x2-x1). In this equation, m is the slope, x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point.

Using this formula, we can solve for b.

m = (y2 - y1) / (x2 - x1)

8/3 = (b - (-1)) / (2 - (1/2))

24/3 = (b + 1) / (5/2)

24/3 * (5/2) = (b + 1)

40/3 = b + 1

40/3 - 1 = b

b = 13/3

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The area of the shaded part of the rectangle is 152 cm².

The area of the shaded part is the difference between the area of the larger rectangle and the smaller square.

Area of the shaded part = area of the larger rectangle - area of the square

Area of the larger rectangle = length x width

12cm x 18cm = 216 cm²

Area of the square = length²

= 8² = 64 cm²

Area of the shaded part = 216 - 64 = 152 cm²

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Mrs. Byrne mowed 1/4 of her lawn. Her son mowed 2/7 of it.  Mrs. Byrne mowed most of the lawn. The lawn still needs to be mowed is 13/28.

Mrs. Byrne mowed of his lawn = 2/7

Her son mowed of his lawn = 1/4

We firstly equal the denominator of both fraction by taking the LCM of both numbers.

The LCM of 7 and 4 is 28.

So we multiply and divide by 4 in the fraction 2/7 and by 7 in fraction 1/4. Now,

Mrs. Byrne mowed of his lawn = 2/7 × 4/4 = 8/28

Her son mowed of his lawn = 1/4 × 7/7 = 7/28

Now we compare the both fraction 8/28 and 7/28. The 8/28 is greater than 7/28. So we can say that Mrs. Byrne mowed most of his lawn.

The total lawn is 28/28.

So, the remaining lawn for mowed = 28/28 - (8/28 + 7/28)

The remaining lawn for mowed = 28/28 - (8 + 7)/28

The remaining lawn for mowed = 28/28 - 15/28

The remaining lawn for mowed = (28 - 15)/28

The remaining lawn for mowed = 13/28

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The complete question is:

Mrs. Byrne mowed 2/7 of his lawn. Her son mowed 1/4 of it. Who mowed most? How much of the lawn still needs to be mowed?

The expression of (²/₃)⁻¹ without an exponent is given as;

3/2

The different laws of exponents are;

1) Power Law of Exponents which is expressed as; (x^m)ⁿ = x^(mn)

2) Quotient Law of Exponents is expressed as; x^m ÷ x^n = x^(m - n)

3) Product Law of Exponents is expressed as; x^(m) * x^(n) = x^(m + n)

4) Law of Reciprocal is expressed as; a⁻ⁿ = 1/aⁿ

Now, we want to rewrite the expression (²/₃)⁻¹ without using an exponent.

From the law of reciprocal of exponents we can say that;

(²/₃)⁻¹ = 1/(²/₃)

= 3/2

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A sample of size 40 is a valid sample size and could result in a sample statistic, as explained above, the correct answer is B.

What is the sample statistic?

A sample statistic is a measure calculated from a sample of the population, while a population parameter is a measure calculated from the entire population. Therefore, a sample statistic may differ from a population parameter due to sampling variability.

Out of the options provided, only option B (40) could result in a sample statistic but not a parameter.

This is because a sample of size 40 is a subset of the population, and a statistic calculated from this sample (such as the sample mean or sample proportion) would be a sample statistic.

Option C (4000) would result in both a sample statistic and a population parameter because a sample consisting of the entire population is a census, and any measure calculated from this sample would also be a parameter.

Option A (both 40 and 4000) and option D (neither 40 nor 4000) are not correct because 4000 is not a valid sample size, as it includes the entire population and would therefore be a census.

Hence, a sample of size 40 is a valid sample size and could result in a sample statistic, as explained above, the correct answer is B.

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The answer is 4^2. Step-by-step explanation: This is because 4 squared is 4*4 which equals 16.

On solving the linear equations x + 3y = 5 and -3x - 2y = 20, the value for x is x = -10 and the value of y is y = 5. When solved independently the value is obtained as -2x + y = 25.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The first linear equation is - x + 3y = 5

The second linear equation is - -3x - 2y = 20

On evaluating individually the equation obtained is -

-3x - 2y + x + 3y = 20 + 5

-2x + y = 25

Multiply equation (1) with 3 -

3(x + 3y) = 5 × 3

3x + 9y = 15.....(3)

Adding equation (3) and (2) -

-3x - 2y + (3x + 9y) = 20 + 15

-3x - 2y + 3x + 9y = 35

-2y + 9y = 35

7y = 35

y = 35/7

y = 5

Substituting the value of y in equation (1) -

x + 3y = 5

x + 3(5) = 5

x + 15 = 5

x = 5 - 15

x = -10

Therefore, the value of x and y is -10 and 5 respectively.

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The Probability that the player will win against each of the four opponents at least once is 0.7942.

The player has a P(W) = 0.80 winning chance.

The likelihood that the player will lose is hence P(L)= 1 - P(W) = 0.20.

The player is up against 4 different opponents in the video game.

It is stipulated that the game finishes when a player is defeated by an opponent.

The player can win in any of the following ways: L, WL, WWL, WWWL, and WWWW.

The results from each of the four opponents are distinct, meaning that the outcome of a game played against one opponent has no bearing on the outcome of a game against another.

P (Player defeats all 4 opponents) = 0.4096 is the probability that a player will win a game against all four opponents.

Thus, the probability that the player defeats all four opponents in a game is 0.4096.

(b) The likelihood that a player will win a game against at least two opponents is,

P = 1 (Player defeats at least 2) P = 0.64 when the player loses the first and second games.

Therefore, there is a 0.64 percent chance that the player will win at least two games against opponents.

Let X equal the number of times the player triumphs over all four opponents.

The likelihood that a player will win a game against all four opponents is,

P(WWWW) = 0.4096

The probability that a player will triumph over each of the four opponents at least once is,

P (X ≥ 1) = 1 - P (X < 1)

= 1 - P (X = 0)\s = 0.7942

Therefore, there is a 0.7942 probability that the player will win against all four opponents at least once.

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Their ages are given below:

The sum of their ages would be 57 years.

The current age of Nico is 27 - 7 = 20 years.

Therefore, Maria and Luis's current ages would be 8 years older than when they were 10 and 11 respectively,

so Maria is currently 18 years old and Luis is currently 19 years old.

Therefore, the sum of their current ages is 18 + 19 + 20 = 57 years.

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8 years ago, Maria was 10 years old and Luis was 11 years old. In 7 years Nico will be 27 years old, what is the sum of the current ages of Maria, Luis and Nico?

Answer:

An exponential function has the form y = ab^x, where a is the initial value, b is the base of the exponential function (in this case, 1 + the growth rate as a decimal) and x is the number of years.

Given that the initial population is 430 and the annual rate of increase is 9%, we can write the exponential function as:

y = 430 (1 + 0.09)^x

Where y is the quail population after x years, and the base (1 + 0.09) represents the growth rate of 9%. This function gives the population of the quail for any number of years x.

It's worth noting that the function is in the form of y = a*(1+r)^x where a is the initial value, r is the annual growth rate, and x is the number of years

The point where the two lines intersect each other is (2,5).

What are simultaneous equations?

Two or more algebraic equations that share variables, such as x and y, are said to be simultaneous equations. Since the equations are solved simultaneously, they are known as simultaneous equations. These equations alone could have an endless number of solutions.

When two lines intersect each other, the point of intersection is a point common to both lines.

This can be found by solving simultaneous equations.

The given equations of the two lines are

3x+y = 11

4x-y = 3

Adding the above equations, we can remove the y variable.

The result is 7x = 14

x = 2

Now we substitute this value of x in either of the above equations to get the value of y.

3 * 2 + y = 11

y = 11 -6 = 5

Therefore the point where the two lines intersect each other is (2,5).

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(3,2) the eqaition would be y=3/2x+3

The equation Wendy must have solve is  14+2x = 54 - 6x Option D

A linear equation is an equation whose highest power of the variable is 1.

recall that equation is a mathematical statement showing that two opposite sides are equal

the equation can be solved as follows

14+2x = 54 - 6x

Collecting the like terms from the equation we have

2x+6x = 54 -14

This implies that

8x = 40

Making x the subject of the relation

Therefore x=40/8 = 5

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Answer:

The new printer prints 100 pages about 4 minutes faster than the old printer

The new printer prints 22 move pages per minute than the old printer

Step-by-step explanation:

New printer's equation

y = 37x

Old printer's equation

y = 15x  

Since the graph is proportional you can take any ordered pair (except (0,0) and put it the form y/x to find the slope. For example, The point (2,30) 30/2 = 15 or the point (4, 60) 60/4 = 15 and so on.

The difference between the rates is 22 pages a minute (37 - 15).

Substituted in 100 for y and solve for x (minutes)

New:

y = 37x

100 = 37x  Divide both sides by 37

100/37 =37x/37

3 = x  This is rounded

This means that it takes about 3 minutes to print 100 pages.

Old:

y = 15x

100 = 15x  Divide both sides by 15

100/15 = 15x/15

7 = x   This is rounded

This means that it takes about 7 minutes to print 100 pages.

The new printer can print the same amount of page (100) in 4 less minutes (7-3)

The correct answer is Between 157,500 and 968,187 phones.

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Answer: 40 cm

Step-by-step explanation: 2/5 x 100 =

200/5 = 40

Answer: To find the values that make the inequality -v <= -8 true, we can first isolate v on one side of the inequality. We do this by adding v to both sides:

-v <= -8

v >= 8

This inequality states that the only values that make it true are those greater than or equal to 8.

Another equivalent inequality in terms of v would be v>=8

So, the values that make the inequality -v <= -8 true are v >= 8

Step-by-step explanation:

The Pythagorean identity is proved to be 1 + cot2 = cosec2, which results in sin2 + cos2 = 1.

With the value of sine and the quadrant in which the angle is placed, we can use the Pythagorean identity to get the angle of cosine.

If we are given the cosine value and the quadrant in which the angle is placed, we may similarly compute the angle of sine.

Cosec2 = 1 + cot2

Therefore,

cosec2 = 1/sin2

Tan2 = cos2 / sin2 = cot2 = 1

Hence,

1/sin2 = 1 + cos2 / sin2.

multiplied both sides by sin2 as a result.

1(sin2 ) + (cos2 / sin2 ) = (sin2 / sin2 )

cos2 + sin2 = 1.

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Answer: B. x + 52

Step-by-step explanation:

The unknown number is represented as x.

The phrase "52 more than" means adding 52 to something

From this we can say that the expression is B. x + 52

The functions that show a positive rate of change include the following:

A. y = 2x/3 - 2.

B. y = 7 - (-3x).

In Mathematics, the slope-intercept form of a line can be represented or modeled by using this linear equation:

y = mx + c

Where:

Generally speaking, a function with a positive rate of change represents an increasing function because its numerical value is increasing over time such as in the following increasing function:

y = 2x/3 - 2.

Slope, m = 2/3 (positive slope).

y = 7 - (-3x)

y = 7 + 3x

Slope, m = 3 (positive slope).

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The functions that show a positive rate of change are y = 2/3x - 2 and y = 7 - (-3x).

A positive function is a function that has the values of x that are greater or higher than zero on the graph. It means that they are always increasing on the positive y-axis i.e. f(x) > 0.

The function's outputs known as the y-values, however, must be larger than zero even though the domain values i.e. (x-values) might be negative. A positive function, then, has values that are positive for all of its domain's parameters.

To determine the positivity of the function on the graph, the slope must also be positive. Using the linear function form y = mx + b.

where;

From the given information, the functions that demonstrated a positive slope are:

Therefore, we can conclude that the functions that show a positive rate of change are y = 2/3x - 2 and y = 7 - (-3x).

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Answer:

Step-by-step explanation:

By applying the proportional relationship formula, the height of the barn is 45 ft.

In a proportional relationship, each pair of data points is related in the same way, typically by multiplying them together. A proportionate relationship can be seen in a set of numbers, an equation, or a graphical representation. In this case, we are given that :

The height of the farmer (X1)  = 6 ft

The height of the farmer’s shadow (Y1) = 10 ft

The height of the barn with farmer’s shadow (Y2) = 75 ft

To calculate the actual height of the barn (X2), we can use this following formula:

X1 : Y1 = X2 : Y2

6 : 10 = X2 : 75

X2 = (6 : 10) x 75

X2 = 45

Thus, the height of the barn is 45 ft

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Answer: 1.365 × 10-6

Step-by-step explanation:

All the numbers in scientific notation written in the form a × 10b, where b is an integer and the coefficient a is a non-zero real number between 1 and 10 in absolute value.

To convert 0.000001361 into scientific notation also known as standard form, follow these steps:

Move the decimal 6 times to right in the number so that the resulting number, m = 1.361, is greater than or equal to 1 but less than 10

Since we moved the decimal to the right the exponent n is negative

n = -6

Write in the scientific notation form, m × 10n

= 1.361 × 10-6

Therefore, 0.000001361 in scientific notation is 1.361 × 10-6 and It has 4 significant figures. In scientific e-notation it is written as 1.361e-6.

The mixed fraction in the simplest form is 21/20, and the snowfall in December was almost 5 times more than the snowfall in January.

A mixed fraction is one that is represented by both its quotient and remainder. A mixed fraction is, for instance, 2 1/3, where 2 seems to be the quotient and 1 is the remainder. An amalgam of a whole integer and a legal fraction is a mixed fraction.

Given that the amount of snowfall in January was:

[tex]1 \frac{1}{20}[/tex]

To convert the mixed fraction to simplest form, multiply the denominator with the value and add it to the numerator:

[tex]21/20 = 1.05[/tex]

The amount of snowfall in December is:

[tex]5 \frac{2}{24} = 5 \frac{1}{12}[/tex]

To convert the mixed fraction to simplest form, multiply the denominator with the value and add it to the numerator:

[tex]\frac{61}{12} = 5.08[/tex]

Hence, the snowfall in December was almost 5 times more than the snowfall in January.

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The equation which represents Lisa's earnings, e, relative to the number of hours, h, that she works assume a proportional relationship is e = 7.5h

Amount Lisa makes for babysitting = $30

Number of hours Lisa babysit = 4 hours

Total earnings = e

Number of hours = h

If the relationship is proportional, then

k = constant of proportionality

e = k × h

30 = k × 4

30 = 4k

divide both sides by 4

k = 30/4

k = 7.5

So therefore,

e = k × h

e = 7.5 × h

e = 7.5h

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Answer:

Quadratic

Step-by-step explanation:

The path of a firework is modeled by a quadratic function. The firework ascends from point of launch with decreasing speed (negative acceleration due to gravity) until it reaches it maximum height and then starts dropping back to ground at increasing speeds due to positive acceleration exerted by the force of gravity.

The flight path of the parabola can thus be modeled by height as a function of elapsed time

h = f(t) where f(t) is a quadratic equation in t with the general form being

a · t² +b · t  + c

We can easily eliminate the linear model since that implies the firework will ascend at a constant speed forever

The exponential model indicates that at the moment of launch the speed of the firework is constant and then it suddenly accelerates. However we know from observation that the firework has the highest speed at the moment of launch

Do not be confused by the shape of the graph - it seems to indicate the firework shoots down and up.

The note says:
Note: The path may be going in a different direction, perhaps reflected or translated vertically or horizontally while maintaining the same shape.

Indeed the actual path of the firework is a reflection of the graph about the x axis. so that it is a downward facing parabola. The vertex of the parabola is the highest y-value which is the maximum height the firework would reach.

The height would be the y axis and time t the x-axis