Write the first five terms of the sequence defined by the explicit formula an=(-2)^n-1

Answer:

Step-by-step explanation:

Given

Find \(a_{43}\)

We know that

Appy this to the two known terms and find the first term and the common difference

Find the coomon difference

Find the first term

Now find the 43th term

Answer:

-98

Step-by-step explanation:

General form of an arithmetic sequence:

  \(a_n=a+(n-1)d\)

where:

Given:

To find the 43rd term we need to construct an equation for the nth term.  To do this, find the first term and the common difference by substituting the given information into the general formula to create a system of equations that can be solved.

\(\begin{aligned} \underline{\textsf{Equation 1}}} \\ a_{21} & =-32 \\ \implies a+(20-1)d & = -32 \\a+20d & =-32 \end{aligned}\)

\(\begin{aligned} \underline{\textsf{Equation 2}}} \\ a_{50} & =-119 \\ \implies a+(50-1)d & = -119 \\a+49d & =-119 \end{aligned}\)

Subtract Equation 1 from Equation 2 to eliminate a:

\(\begin{array}{r l}a+49d & = -119\\- \quad a+20d & = -32\\\cline{1-2}29d & = -87\end{aligned}\)

Solve for d:

\(\implies d=\dfrac{-87}{29}=-3\)

Substitute the found value of d into one of the equations and solve for a:

\(\implies a + 20(-3)=-32\)

\(\implies a=28\)

Substitute the found values of a and d into the general formula to create an equation for the nth term:

\(\implies a_n=28+(n-1)(-3)\)

\(\implies a_n=31-3n\)

Finally, substitute n = 43 into the found formula to find the 43rd term of the sequence:

\(\implies a_{43}=31-3(43)=-98\)

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Thus, the output of the loop is non-deterministic.

Let us consider the loop and the conditions given:

While the condition `x ≠ 0` holds, the loop will continue.

Within the loop, there are two possible paths: `y:=y1` or `y:=y2`.

Since the path to take is non-deterministic, there are two possible outputs:

Either y will be assigned a value of y1 or y2.

Here, it is impossible to determine which assignment will be executed; there is no way to know if x will be decremented once or twice.

This demonstrates that the behavior is non-deterministic, and the possible outputs are `[y1,y2]`.

Thus, the output of the loop is non-deterministic.

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Answer:

The temperature changed by 7°F each day.

Step-by-step explanation:

The temperature changed a total of 42 degrees throughout the six days, dividing it into 7 degrees for each day.

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Answer:

41

Step-by-step explanation:

b^2 -2ac

Let  a=4  b=3  c=-4

( 3)^2 -2 (4) (-4)

Exponents first

9 - 2 (4) (-4)

Multiply

9 + 32

41

Answer:

41

Step-by-step explanation:

b^2 - 2ac                plug in the values

3^2 -2(4)(-4)             start with the exponent (remember PEMDAS)

9  -2(4)(-4)                start multiplying from left to right

9  -8(-4)                    multiply

9 + 32                       add

41                              that's your answer :)  

Answer:

parallel

Step-by-step explanation:

paralell

To solve the equation 3a² = -4a + 15 by factoring, we need to rewrite it in the form of a quadratic equation, set it equal to zero, and then factor it. The solutions to the equation 3a² = -4a + 15 are a = 5/3 and a = -3.

The equation 3a² = -4a + 15 can be rearranged as 3a² + 4a - 15 = 0. Now we can factor the quadratic expression.

To factor the quadratic expression, we need to find two numbers that multiply to give -45 and add up to +4. The numbers that satisfy these conditions are +9 and -5. So, we can write the equation as (3a - 5)(a + 3) = 0.

Setting each factor equal to zero, we have two possible solutions: 3a - 5 = 0 or a + 3 = 0.

Solving these equations, we find a = 5/3 or a = -3.

Therefore, the solutions to the equation 3a² = -4a + 15 are a = 5/3 and a = -3.

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Answer:

•1/6

Step-by-step explanation:

the sides of the original is 6 by 6 , the sides of the copy is 1 by 1. therefore it is a fraction of 1/6

Answer:

10.8 inches

Step-by-step explanation:

72/20=3.6

3.6*3=10.8

The value of x that makes the equation true is 8

A linear equation is a type of equation that the highest power of it's variable is 1. Example of linear equation include, x+2 =y, x+7/2 =15.

The graph of a linear equation is a straight line and it is represented in form of y = mx+b. Where m is the slope and b is the y intercept.

Solving x-7 = (3/4)x -5

collecting like terms

x-(3/4)x = 7-5

x/4 = 2

x = 8

therefore the value of x that will make the equation x-7 = (3/4)x -5 true is 8.

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Answer:

14.75789056

Step-by-step explanation:

I think so...

The first five terms of the binomial theorem to approximate (-1.4)⁸ answer would be 14.056.

The binomial theorem states the principle for expanding the algebraic expression (x + y)ⁿ and describes it as the total of the phrases involving the unique exponents of the x and y variables. Each word in a binomial expansion has a coefficient, which is a numerical value.

According to the binomial theorem, it is possible to expand any non-negative power of binomial (x + y) into a sum of the form,

(x+y)ⁿ = ⁿC₀xⁿy₀ + nC₁ xⁿ⁻¹y₁ + nC₂ xⁿ⁻² y2 + ... + nC x¹yⁿ⁻¹ + nCn x°yⁿ

where n ≥ 0 is an integer and each ⁿCₓ is a positive integer known as a binomial coefficient.

Given expression as (- 1.4)⁸

as a result, if you have raised a negative integer to an even power, the response will be positive.

Eliminating the negative sign.

⇒ (1.4)⁸ = (1+0.4)⁸

⇒ 1⁸+8*1⁷*0.4+28*1⁶*0.4²+56*1⁵*0.4³+70*1⁴*0.4⁴

⇒ 1+3.2+4.48+3.584+1.792

⇒ 14.056

If we used one more term we would get 14.62944

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The values of  θ for the given inequality be ⇒ 3π/4 < θ < π

To determine the values of θ for which

cosθ < sinθ  for 0 ≤ x < π,

Now use the trigonometric identity,

sin²(θ) + cos²(θ) = 1

Rearranging this equation:

sin²θ = 1 - cos²θ

Then,

Substitute this in the original inequality, we get

⇒ cosθ < sinθ

⇒ cosθ < √(1 - cos²θ)

Squaring both sides:

⇒ cos²θ< 1 - cosθ

⇒ 2cos²θ < 1

Taking the square root:

cosθ < √(1/2)

cosθ < √(2)/2

So, the solution is:

0 ≤θ < π/4   or   3π/4 < θ < π

Hence,

3π/4 < θ < π is the solution.

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The value of k in the rational expression (5k - 2)/3k = 7 is - 2/17.

A mathematical statement expressed as a string of numbers and unknowable variables is known as a numerical expression. Statements can be used to create numerical expressions.

We also know that a fraction is written in the form of p/q, where q ≠ 0.

Given, A rational expression (5k - 2)/3k = 7.

Now, The denominator of the LHS will be multiplied to the numerator of RHS.

Therefore, 5k - 2 = 21k.

5k - 21k = 2.

- 17k = 2.

- k = 2/17.

k = - 2/17.

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The process capability ratio is approximately 0.67 (option A).

To calculate the process capability ratio, we need to use the following formula:

Process Capability Ratio = (Upper Specification Limit - Lower Specification Limit) / (6 * Process Standard Deviation)

Given the following values:

Upper Specification Limit = Diameter + Tolerance = 6.77 cm + 0.01 cm = 6.78 cm

Lower Specification Limit = Diameter - Tolerance = 6.77 cm - 0.01 cm = 6.76 cm

Process Standard Deviation = 0.005 cm

Plugging these values into the formula, we have:

Process Capability Ratio = (6.78 cm - 6.76 cm) / (6 * 0.005 cm)

Process Capability Ratio = 0.02 cm / 0.03 cm

Process Capability Ratio ≈ 0.67

Therefore, the process capability ratio is approximately 0.67. The correct answer is A) 0.67.

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The statement "a. The slope is negative" is true about the slope of the least squares regression line when the correlation coefficient is negative.

When the correlation coefficient is negative, it indicates an inverse relationship between the two variables. In a linear regression, the slope of the line represents the direction and magnitude of the relationship between the independent and dependent variables. A negative correlation coefficient indicates that as the independent variable increases, the dependent variable decreases. Therefore, the slope of the least squares regression line will also be negative.

The slope of the regression line is calculated using the formula: slope = correlation coefficient * (standard deviation of y / standard deviation of x). Since the correlation coefficient is negative and the standard deviation of x and y are positive values, multiplying a negative correlation coefficient by positive standard deviations will result in a negative slope. Hence, option "a. The slope is negative" is the correct statement.

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Answer:

Graph B

Step-by-step explanation:

The vertical intercept I am getting is (0,-1) and B is the only one with that point. Therefore, it should be B.

We are 99% confident that the true average stance duration among elderly individuals lies within the range of 744.56 ms to 857.44 ms.

To test whether the true average stance duration is larger among elderly individuals than among younger individuals, we can perform a one-tailed independent samples t-test. The null hypothesis (H0)

Using the t-test, we compare the means and standard deviations of the two samples and calculate the test statistic

a) To calculate a 99% confidence interval for the true average stance duration among elderly individuals, we can use the sample mean, sample standard deviation, and the t-distribution.

Given:

Older adults: n = 28, sample mean = 801, sample standard deviation = 117

Using the formula for a confidence interval for the mean, we have:

Margin of error = t * (sample standard deviation / √n)

Since the sample size is relatively large (n > 30), we can use the z-score instead of the t-score for a 99% confidence interval. The critical z-value for a 99% confidence level is approximately 2.576.

Calculating the margin of error:

Margin of error = 2.576 * (117 / √28) ≈ 56.44

The confidence interval is then calculated as:

Confidence interval = (sample mean - margin of error, sample mean + margin of error)

Confidence interval = (801 - 56.44, 801 + 56.44) ≈ (744.56, 857.44)

b) To test whether the true average stance duration is larger among elderly individuals than among younger individuals, we can perform a one-tailed independent samples t-test.

The null hypothesis (H0): The true average stance duration among elderly individuals is equal to or less than the true average stance duration among younger individuals.

The alternative hypothesis (Ha): The true average stance duration among elderly individuals is larger than the true average stance duration among younger individuals.

. With the given data, perform the t-test and obtain the p-value.

c) To construct a 95% confidence interval for the difference in means between older and younger adults, we can use the formula for the confidence interval of the difference in means.

Given:

Older adults: n1 = 28, sample mean1 = 801, sample standard deviation1 = 117

Younger adults: n2 = 16, sample mean2 = 780, sample standard deviation2 = 72

Calculating the standard error of the difference in means:

Standard error = √((s1^2 / n1) + (s2^2 / n2))

Standard error = √((117^2 / 28) + (72^2 / 16)) ≈ 33.89

Using the t-distribution and a 95% confidence level, the critical t-value (with degrees of freedom = n1 + n2 - 2) is approximately 2.048.

Calculating the margin of error:

Margin of error = t * standard error

Margin of error = 2.048 * 33.89 ≈ 69.29

The confidence interval is then calculated as:

Confidence interval = (mean1 - mean2 - margin of error, mean1 - mean2 + margin of error)

Confidence interval = (801 - 780 - 69.29, 801 - 780 + 69.29) ≈ (-48.29, 38.29)

Comparison with part (b): In part (b), we performed a one-tailed test to determine if the true average stance duration among elderly individuals is larger than among younger individuals. In part (c), the 95% confidence interval for the difference in means (-48.29, 38.29) includes zero. This suggests that we do not have sufficient evidence to conclude that the true average stance duration is significantly larger among elderly individuals compared to younger individuals at the 95% confidence level.

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To paint the circular table top Beth need to buy 2 cans of paint .

In the question ,

it is given that ,

the radius of the circular table top is = 3 feet

So , the area of the circular table top is = π*r² = π*(3)²

= 9π

= 28.26

also given that

area covered by 1 small can of paint = 15 sq. feet

So , the number of cans needed to paint the table top is = (area of table top)/(area covered by 1 small can of paint)

On substituting the values , we get

number of cans needed to paint the table top is = 28.26/15

= 1.884

≈ 2

Therefore , To paint the circular table top Beth need to buy 2 cans of paint  .

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The first statement says that if we multiply two positive real numbers and get a result larger than 400, then at least one of the two numbers must be greater than 20 Second statement talks about the sum of two positive real numbers being greater than 400

Similarly, the second statement talks about the sum of two positive real numbers being greater than 400. In this case, if both numbers were less than or equal to 200, their sum would be less than or equal to 400. Thus, for the sum to be greater than 400, at least one of the two numbers must be greater than 200.

These statements are important in solving problems related to real-life scenarios, such as finding the dimensions of a room or the possible values of a variable in an equation. They help us identify the minimum value of a variable or the minimum condition that must be met to achieve a certain result.

In conclusion, the two statements show us that the multiplication and addition of real numbers have certain conditions that must be met to obtain a specific result. They are useful in problem-solving and understanding mathematical concepts.

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Complete question is

"a. If the product of two positive real numbers is larger than 400, then at least one of the two numbers is larger than 20.

b. If the sum of two positive real numbers is larger than 400, then at least one of the two numbers is larger than 200."

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Explanation:

5x+y = 31 solves to y = -5x+31 after subtracting 5x from both sides.

We plug this into the other equation to get....

-3x + 4y = -37

-3x + 4( y ) = -37

-3x + 4( -5x+31 ) = -37 .... y replaced with -5x+31

-3x - 20x + 124 = -37 ... distribute

-23x+124 = -37

-23x = -37-124 .... subtract 124 from both sides

-23x = -161

x = -161/(-23) .... divide both sides by -23

x = 7

Use this x value to find y

y = -5x+31

y = -5(7)+31 ... plug in x = 7

y = -35+31

y = -4

The solution as an ordered pair is (x,y) = (7, -4)

If you graphed the original equations on the same xy grid, you should find they cross or intersect at (7, -4).

The least positive number n that satisfies all of the conditions is n = 60.

To find the least positive number n that satisfies all of these conditions, we need to find the least common multiple (LCM) of the numbers 1, 2, 3, 4, and 5. The LCM is the smallest positive number that is divisible by all of these numbers.

To find the LCM, we can list the prime factors of each number and then take the highest power of each prime factor that appears in any of the numbers. The prime factorization of each number is:

1: 1

2: 2

3: 3

4: 2 x 2

5: 5

The highest power of each prime factor that appears in any of the numbers is:

2: 2 x 2

3: 3

5: 5

So the LCM is:

\(2 \times 2 \times 3 \times 5$ = 60\)

Therefore, the least positive number n that satisfies all of the conditions is n = 60.

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Answer: 17.5

Step-by-step explanation:

7:10 = x:25

times both sides by 25

25(7/10) =x

than solve for x

X = 25 × (7/10)

X = 17.5

Answer:

48 cups

Step-by-step explanation:

There are 16 cups in one gallon, and 16 x 3 equals 48

Answer:

b

Step-by-step explanation:

15/6=2.5

100/2.5=40

Answer:

5x+2y<=30

Step-by-step explanation:

Inequalities help us to compare two unequal expressions. The graph for the inequality can be made as shown below.

Inequalities help us to compare two unequal expressions. Also, it helps us to compare the non-equal expressions so that an equation can be formed. It is mostly denoted by the symbol <, >, ≤, and ≥.

As per the graph let the number of pounds of quinoa be represented by x and the number of pounds of rice Celeste can buy be represented by y.

Therefore, the expression that can represent the number of pounds that Celeste can buy is,

5x + 2y

Also, it is given that Celeste can spend no more than $30. Therefore, the inequality can be written as,

5x + 2y ≤ 30

Now the graph for the inequality can be made as shown below.

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Answer:

A. Trapezoid

B. Isoclese triangle

c. Equilateral triangle

d. Kite

Answer:

A, Parallelogram (more specifically a trapezoid)

B. Isosceles Triangle

C. Equilateral Triangle

D. Kite

Answer:

  C.  142°

Step-by-step explanation:

You want the angle between vectors u=3i+√3j and v=-2i-5j.

There are a number of ways the angle between the vectors can be found. For example, the dot-product relation can give you the cosine of the angle:

  u•v = |u|·|v|·cos(θ) . . . . . . where θ is the angle of interest

You can find the angles of the vectors individually, and subtract those:

  u = |u|∠α

  v = |v|∠β

  θ = α - β

When the vectors are expressed as complex numbers, the angle between them is the angle of their quotient:

  \(\dfrac{\vec{u}}{\vec{v}}=\dfrac{|\vec{u}|\angle\alpha}{|\vec{v}|\angle\beta}=\dfrac{|\vec{u}|}{|\vec{v}|}\angle(\alpha-\beta)=\dfrac{|\vec{u}|}{|\vec{v}|}\angle\theta\)

This method is used in the calculation shown in the first attachment. The angle between u and v is about 142°.

A graphing program can draw the vectors and measure the angle between them. This is shown in the second attachment.

__

Additional comment

The approach using the quotient of the vectors written as complex numbers is simply computed using a calculator with appropriate complex number functions. There doesn't seem to be any 3D equivalent.

The dot-product relation will work with 3D vectors as well as 2D vectors.

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Answer:

r = 2.4

Step-by-step explanation:

Given, circumference = 15.2 mm

We know that, circumference = 2πr

So, 2πr = 15.2 mm

=> 2(3.14)r = 15.2

=> 3.14r = 15.2/2

=> 3.14r = 7.6

=> r = 7.6/3.14

=> r = 2.42...

=> r = 2.4 [rounded to one decimal place]

Answer:

A

Step-by-step explanation:

When the temperature has a negativ sign on it, it is below zero

When the temperature is a natural number, it is above zero

Answer: above zero

Step-by-step explanation:

Because if it was below zero there would be a negative Infront of the number so show it’s below zero but there’s not, so I’m pretty sure it would be above zero. I hope this helps