F(n) = 2(-3)^n complete the recursive formula of f(n)

Answer:

x = -2 - sqrt(3), -2 + sqrt(3) 

Step-by-step explanation:

We can rewrite the equation as

x² + 4x + 1 = 0

Now we can use the quadratic equation.

x = (-b ± sqrt(b² - 4ac)) / 2a

where a = 1, b = 4, c = 1. Substituting these values ​​gives:

x = (-4 ± sqrt(4² - 4(1)(1))) / 2(1)

x = (-4 ± sqrt(16 - 4)) / 2

x = (-4 ± sqrt(12)) / 2

x = (-4 ± 2sqrt(3)) / 2

x = -2 ± sqrt(3)

So, from min to max, the solution is:

x = -2 - sqrt(3), -2 + sqrt(3) 

Hope this helps!

Answer:

11.5

Step-by-step explanation:

Add 7+8+9+9+11+11+12+14+15+19. Divide it all by 10 (the number of values.)

Answer:

11.5

Step-by-step explanation:

Add 7+8+9+9+11+11+12+14+15+19. Divide it all by 10 (the number of values.)

The area of the pillow is 50 square inches.

To find the area of the pillow shaped as a regular pentagon made from 5 congruent triangles, each with an area of 10 square inches, follow these steps:

1. Identify the number of triangles: There are 5 congruent triangles in the pentagon.

2. Determine the area of each triangle: Each triangle has an area of 10 square inches.

3. Calculate the total area: Multiply the number of triangles (5) by the area of each triangle (10 square inches).

5 triangles * 10 square inches/triangle = 50 square inches

So, the area of the pillow is 50 square inches.

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The following parts can be answered by the concept from Probability.

a. The mean of the sample proportion is also 0.70.

b. The variance of the sample = (0.70(1-0.70))/100 = 0.0021

c. The standard error of the sample is 0.0458

d. The probability that the sample proportion exceeds 0.80 is  2.18

a. The mean of the distribution of the sample proportion of returns leading to refunds can be found using the formula:
mean = p = 0.70, where p is the population proportion of returns leading to refunds.
Therefore, the mean of the sample proportion is also 0.70.

b. The variance of the sample proportion can be found using the formula:
variance = (p(1-p))/n, where n is the sample size.
Substituting the given values, we get:
variance = (0.70(1-0.70))/100 = 0.0021

c. The standard error of the sample proportion can be found using the formula:
standard error = sqrt(variance)
Substituting the calculated variance value, we get:
standard error = √(0.0021) = 0.0458

d. To find the probability that the sample proportion exceeds 0.80, we need to standardize the sample proportion using the formula:
z = (sample proportion - population proportion) / standard error
Substituting the given values, we get:
z = (0.80 - 0.70) / 0.0458 = 2.18

Using a standard normal distribution table or calculator, we can find the probability of getting a z-score of 2.18 or higher, which is approximately 0.015 or 1.5%. Therefore, the probability that the sample proportion exceeds 0.80 is 0.015 or 1.5%.

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The derivative of the function y = sec⁻¹(4s³ + 9) is [tex]dy/ds = (12s^2) / (|4s^3 + 9| * \sqrt{((4s^3 + 9)^2 - 1))}[/tex].

We have to find the derivative of y with respect to s for the given function y = sec⁻¹(4s³ + 9).

Here are the steps to find the derivative:

1. Identify the function:

y = sec⁻¹(4s³ + 9).

2. Apply the chain rule:

dy/ds = (dy/du) * (du/ds), where u = 4s³ + 9.

3. Find dy/du:

Since y = sec⁻¹(u), the derivative

[tex]dy/du = 1 / (|u| * \sqrt{(u^2 - 1)}).[/tex]

4. Find du/ds:

Since u = 4s³ + 9, the derivative du/ds = 12s².

5. Combine the derivatives:

[tex]dy/ds = (1 / (|4s^3 + 9| * \sqrt{((4s^3 + 9)^2 - 1))}) * (12s^2)[/tex].

So, the derivative of y with respect to s for the function y = sec⁻¹(4s³ + 9) is:

[tex]dy/ds = (12s^2) / (|4s^3 + 9| * \sqrt{((4s^3 + 9)^2 - 1))}[/tex]

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Step-by-step explanation:

Volume of a cylinder = pi r^2 h     <=====solve for 'h'

h = volume / (pi r^2)

  = 348.3 mm^3 / ( pi * 4.5^2)             ( I assumed the dimension mm^3 )

h = ~ 5.5 mm

Answer:

Yes, these two figures are congruent. Rotate the figure on the left 90° counterclockwise, and it will look just like the figure on the right.

If f(6)=14 f' is continuous and f'(x)dx=18 the value of f(7) is 32.

To find the value of f(7), we need to use the fundamental theorem of calculus, which states that if f is a continuous function and f'(x) is its derivative, then:

∫f'(x)dx = f(x) + C

where C is the constant of integration.

Given that f' is continuous and f'(x)dx=18, we can integrate both sides to obtain:

∫f'(x)dx = ∫18 dx

Using the fundamental theorem of calculus, we get:

f(x) + C = 18x + K

where K is another constant of integration.

Now, we can use the given value of f(6) to solve for C. Since f(6) = 14, we have:

f(6) + C = 18(6) + K

14 + C = 108 + K

C - K = 94

Substituting this value of C into our equation, we get:

f(x) = 18x + K - 94

To find the value of f(7), we substitute x = 7 into this equation:

f(7) = 18(7) + K - 94

Simplifying, we get:

f(7) = 100 + K

Therefore, we need to find the value of K to determine f(7). We can use the given information that f' is continuous to conclude that f is differentiable. Thus, we can differentiate our equation for f(x) to obtain:

f'(x) = 18

Since f'(x) is constant, we know that f(x) is a linear function of x. Therefore, we can use the two given points (6, 14) and (7, f(7)) to solve for K. The slope of the line passing through these points is:

m = (f(7) - 14) / (7 - 6) = f(7) - 14

Solving for f(7), we get:

f(7) - 14 = 18

f(7) = 32

Therefore, the value of f(7) is 32.

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The probability that the series goes to 7 games is approximately 0.2734, or 27.34%.To find the probability that the series goes to 7 games, we can use the binomial distribution. Let X be the random variable representing the number of games won by one of the teams in a best-of-seven series.

Then, X follows a binomial distribution with parameters n=7 and p=0.5, where n is the number of trials & p is the probability of success in each trial (i.e., winning a game).

Both sides must win three games apiece in the first six games for the series to proceed to seven. The series winner will then be decided in the seventh game.

As a result, the likelihood that the series will go to 7 games is the same as the likelihood that each side will win precisely 3 of the first 6 games, which is:

P(X=3) = (7 choose 3) * (0.5)^3 * (1-0.5)^(7-3) = 35/128 = 0.2734

where (7 choose 3) is the number of ways to choose 3 games out of 7. Therefore, the probability that the series goes to 7 games is approximately 0.2734, or 27.34%.

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(a) The value of S3 = (13/2 - 13/3√2 + 13/4√3) ≈ 6.695

(b) The value of using S3 is Sn = 13(1 - 1/2√2 + 1/3√3 - ... - 1/(n+1)√(n+1))

(c) The series ∑n=1 an converges, and its value is approximately 25.506.

In part (a), we are given a sequence an and asked to find S3, which is the sum of the first three terms of the sequence. We substitute n=1, 2, and 3 in the formula for an and add the resulting values to get S3 = -194.67.

In part (b), we are asked to find a simplified formula for Sn, which is the sum of the first n terms of the sequence. We notice that an can be written as 13 times the difference between two terms involving square roots of (n+1) and n. Using algebraic manipulation, we obtain Sn = 13[(1/√2) - (1/√{n+1})], which simplifies to Sn = 13/√2 - 13/√{n+1}.

In part (c), we use the formula obtained in part (b) to find the sum of the infinite series ∑n=1 an. As n approaches infinity, the second term in the formula approaches zero, so the sum approaches 13/√2. Therefore, the sum converges to a finite value of approximately 9.19.

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The area under the standard normal curve between z = -1.97 and z = -0.79 is approximately 0.1904, rounded to four decimal places.

To find the area under the standard normal curve between z = -1.97 and z = -0.79, you will need to use a Z-table or a calculator with a normal distribution function.

1. Find the area to the left of z = -1.97 and z = -0.79 in the Z-table or using a calculator.
2. Subtract the area of z = -1.97 from the area of z = -0.79 to get the area between the two points.

Using a Z-table or calculator, you will find the areas to the left are:

- For z = -1.97, the area is 0.0244
- For z = -0.79, the area is 0.2148

Now, subtract the smaller area from the larger area:

0.2148 - 0.0244 = 0.1904

So, the area under the standard normal curve between z = -1.97 and z = -0.79 is approximately 0.1904, rounded to four decimal places.

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The correct answer is c.

The interpretation of the coefficient $0.03 in the regression of textbook retail price (PRICE) on number of pages in the book (LENGTH) is: As the length of the book increases by one page, the estimated price increases by $0.03 on average. Therefore, the correct answer is c.

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The conditions under which the K-means algorithm will coverage or end are: A. All the data points have their own cluster; B. No centroids need to move their location; C. No data points need to change their cluster; D. All clusters have sufficient data points; E. The clustering yields the desirable number of clusters.

K-means algorithm need multiple iterations to generate desirable results.

The conditions under which the K-means algorithm will coverage or end are s follows:

A - If all data points have their own cluster, the algorithm has covered all data points and there is no need for further iterations.
B - If no centroids need to move their location, it means that they have already converged to the optimal position and further iterations are not necessary.
C - If no data points need to change their cluster, it means that the clusters have already been formed optimally and further iterations are not needed.
E - If the algorithm has generated the desirable number of clusters, there is no need for further iterations.

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The correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

To evaluate the integral f'(x) f(x) dx using substitution, we can let u = f(x), so that du/dx = f'(x) and dx = du/f'(x). Substituting these expressions into the integral, we get:

∫ f'(x) f(x) dx = ∫ u du

Integrating u with respect to itself, we get:

∫ u du = (u^2)/2 + C

Substituting back for u, we get:

∫ f'(x) f(x) dx = (f(x)^2)/2 + C

Therefore, the correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

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The correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

To evaluate the integral f'(x) f(x) dx using substitution, we can let u = f(x), so that du/dx = f'(x) and dx = du/f'(x). Substituting these expressions into the integral, we get:

∫ f'(x) f(x) dx = ∫ u du

Integrating u with respect to itself, we get:

∫ u du = (u^2)/2 + C

Substituting back for u, we get:

∫ f'(x) f(x) dx = (f(x)^2)/2 + C

Therefore, the correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

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Answer:

  see attached

Step-by-step explanation:

You want to graph the solution to the inequality 4.5x -100 > 125 on the number line.

The inequality is solved the same way you would solve a 2-step equation:

  4.5x -100 > 125 . . . . . . given

  4.5x > 225 . . . . . . add 100 to both sides to eliminate unwanted constant

  x > 50 . . . . . . . divide both sides by 4.5 to eliminate unwanted coefficient

Values of x that are greater than 50 are to the right of 50 on the number line. An open circle is used at x=50, because x=50 is not part of the solution.

The monthly payment needed to amortize a typical $135,000 mortgage loan amortized over 30 years at an annual interest rate of 6.9% compounded monthly is $849.06.

To find the monthly payment:

The formula to calculate the monthly payment needed to amortize a mortgage loan is:

M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1]

Where:
M = Monthly payment
P = Loan amount (in this case, $135,000)
i = Interest rate per month (6.9% / 12 = 0.575%)
n = Total number of payments (30 years x 12 months per year = 360)

Substituting the values into the formula, we get:

M = $135,000 [ 0.00575(1 + 0.00575)^360 ] / [ (1 + 0.00575)^360 – 1]
M = $849.06

Therefore, the monthly payment needed to amortize a typical $135,000 mortgage loan amortized over 30 years at an annual interest rate of 6.9% compounded monthly is $849.06.

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Using linear approximation, (32.05)^(4/5) is equal to 16.08 (rounded to two decimal places).

To use linear approximation (or differentials) to estimate (32.05)^(4/5), we'll first find the function and its derivative, then choose a nearby value to approximate from.
1. Define the function: f(x) = x^(4/5)
2. Find the derivative: f'(x) = (4/5)x^(-1/5)
Now, let's choose a nearby value that is easy to work with. In this case, we'll choose x=32.
3. Evaluate f(32) and f'(32):
  f(32) = 32^(4/5) = 16
  f'(32) = (4/5)(32)^(-1/5) = (4/5)(2) = 8/5
Now we can use linear approximation:
4. Δx = 32.05 - 32 = 0.05
5. Δf ≈ f'(32) × Δx = (8/5) × 0.05 = 0.08
Lastly, approximate the value:
6. f(32.05) ≈ f(32) + Δf ≈ 16 + 0.08 = 16.08

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The conclusion of the Durbin-Watson test is that the test is inconclusive. Therefore option e is correct.

To determine the conclusion of the Durbin-Watson test:

Follow these steps:

STEP 1: Compare the test statistic (1.58) to the critical values (d_L=0.8 and 2d_U=1.3).
STEP 2: If the test statistic is less than d_L or greater than 4-d_L, reject the null hypothesis and conclude that autocorrelation exists.
STEP 3: If the test statistic is between d_U and 4-d_U, do not reject the null hypothesis and conclude that there is insufficient evidence of autocorrelation.
STEP 4: If the test statistic is between d_L and d_U or between (4-d_U) and (4-d_L), the test is inconclusive.

In this case, the test statistic (1.58) is between d_L (0.8) and 2d_U (1.3).

Therefore, the conclusion of the Durbin-Watson test is that the test is inconclusive (Option e).

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Answer:

Step-by-step explanation:

A and B are equivalent since they both simplify to x³ - x² + 4. Therefore, the answer is A and B.

To determine which polynomials are equivalent, we need to simplify each polynomial first. The simplified forms of the given polynomials are:

A. (x³ - 2x + 1) - x(x - 2) = x³ - x² + 2x - 1

B. x(x²-4) - (x - 2)² = x³ - x² + 4

C. x³-(x - 1)(x + 1) = x³ - (x² - 1) = x³ - x² + 1

D. x(x - 2)² + 3x(x - 1) = x³ - x² + 5x

E. -(2x² + 3) + (x³ + x) + (3x² - x + 2) = x³ + x² - x - 1

From the simplified forms, we can see that polynomials A and B are equivalent since they both simplify to x³ - x² + 4. Therefore, the answer is A and B.

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The total number of meters that Morgan rode in all would be 4, 600 m .

Morgan rode her bike 2 kilometers to her friend's house and then eventually rode back home so the distance rode was ;

= 2 km + 2 km

= 4 km

In meters, this would be:

= 4 km x 1, 000 meters per km

= 4 km x 1, 000

= 4, 000 m

Then, she rode 600 meters to and from the library for a total of :

= 4, 000 + 600

= 4, 600 m

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To implement the given function using one 8x1 multiplexer, we first need to identify the inputs and outputs. The inputs are A, B, C, and D, and the output is F.

We can use the 8x1 multiplexer as a logic function generator by using the select inputs to choose which input is passed to the output.

To implement the given function, we can use the following steps:

1. Connect A and D to the select inputs of the multiplexer.
2. Connect B and C to the remaining two inputs of the multiplexer.
3. Set the outputs of the multiplexer as follows:
- Connect output 0 to VCC.
- Connect output 1 to B.
- Connect output 2 to A.
- Connect output 3 to BC'.
- Connect output 4 to AB.
- Connect output 5 to AC.
- Connect output 6 to B'CD.
- Connect output 7 to ABCD'.

4. Connect the multiplexer outputs to a logical OR gate to generate the final output F.

By setting the select inputs appropriately, the multiplexer will output the required terms of the function, which are then combined using the OR gate to produce the final output F.
Hi! To implement the given function F(A, B, C, D) = A'C'B + AB'C' + B'C'D + ABCD' using one 8x1 multiplexer, follow these steps:

1. Identify the input and control lines: Since it is an 8x1 multiplexer, we need three control lines. Choose A, B, and C as the control lines. The input lines will be connected based on the function.

2. Map the function to the input lines: For an 8x1 multiplexer, the inputs are connected as follows:
 - I0 = A'B'C'D'
 - I1 = A'B'C'D
 - I2 = A'B'CD'
 - I3 = A'B'CD
 - I4 = AB'C'D'
 - I5 = AB'C'D
 - I6 = ABCD'
 - I7 = ABCD

3. Connect the corresponding function terms to the input lines:
 - I0 = 0 (A'B'C'D' does not appear in the function)
 - I1 = A'C'B (A'B'C'D matches the first term)
 - I2 = 0 (A'B'CD' does not appear in the function)
 - I3 = B'C'D (A'B'CD matches the third term)
 - I4 = AB'C' (AB'C'D' matches the second term)
 - I5 = 0 (AB'C'D does not appear in the function)
 - I6 = ABCD' (ABCD' matches the fourth term)
 - I7 = 0 (ABCD does not appear in the function)

By connecting the input lines according to the function terms and using A, B, and C as the control lines, you can implement the given function F(A, B, C, D) using one 8x1 multiplexer.

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The choice of reinforcement schedule can have important implications for both learning and the persistence of behavior over time.

The reinforcement schedule can have a significant impact on how quickly a response/behavior is learned and how quickly extinction occurs.

In general, continuous reinforcement schedules (where the behavior is reinforced every time it occurs) tend to result in faster learning of the behavior than partial reinforcement schedules (where the behavior is only reinforced some of the time). This is because the individual learns more quickly that the behavior is associated with the reinforcement.

However, once the behavior is learned, partial reinforcement schedules tend to result in greater resistance to extinction than continuous reinforcement schedules. This is because the individual has learned that the behavior is not always followed by reinforcement, so they are more likely to persist in the behavior even if reinforcement is no longer provided.

Among partial reinforcement schedules, fixed ratio schedules (where reinforcement is provided after a fixed number of responses) tend to lead to the fastest responding and highest rates of responding, but also tend to result in rapid extinction once reinforcement is removed. In contrast, variable ratio schedules (where reinforcement is provided after an average number of responses, with some variation) tend to lead to more stable responding and slower extinction. Fixed interval and variable interval schedules (where reinforcement is provided after the first response following a fixed or variable amount of time) tend to lead to moderate rates of responding and moderate resistance to extinction.

Overall, the choice of reinforcement schedule can have important implications for both learning and the persistence of behavior over time.

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It will take 24.13 minutes for 20% of the water to leak from the vase when it is full of water.

The volume of a cone is given by the formula:

V = 1/3 πr²h

where r = 4.8/2 = 2.4 inches and h = 10 inches

Volume of vase = 1/3 * 22/7 * 2.4² * 10

Volume of vase = 19.2π in³

20% of volume will be:

20/100 * 19.2π = 3.84π in³

Rate = Volume / time

time = Volume / Rate

time = 3.84π / 0.5

time = 24.13 minutes

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The null hypothesis that the pump's average flow rate is 10 liters/sec cannot be ruled out at the 0.01 level of significance.

A one-sample t-test can be used to determine whether a specific pump's average flow rate is 10 litres per second.

The alternative hypothesis is that the population mean flow rate is not 10 liters/sec, contrary to the null hypothesis that it is.

The test statistic, where the hypothesised mean is 10 liters/sec, is calculated as follows: t = (sample mean - hypothesised mean) / (sample standard deviation / sqrt(sample size)).

First, we must determine the sample mean and sample standard deviation: sample mean = (10.05 liters/sec) sample standard deviation =

10.05 litres per second is the sample mean (10.2 + 9.7 + 10.1 + 10.3 + 10.1 + 9.8 + 9.9 + 10.4 + 10.3 + 9.8)/10.

0.23 litres per second.

The formula for t is given as follows after substituting these values: t = (10.05 - 10) / (0.23 / [tex]\sqrt{10}[/tex]) = 1.3

For this test, n - 1 = 9 represents the degrees of freedom.

The crucial t-value is found to be 3.250 using a t-distribution table with 9 degrees of freedom and a significance threshold of 0.01 (two-tailed).

We are unable to reject the null hypothesis since the calculated t-value (1.3) is less than the crucial t-value (3.250).

Therefore, we lack sufficient data to draw the conclusion that the pump's average flow rate deviates from 10 liters/sec at the 0.01 level of significance.

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Answer:12.375

Step-by-step explanation:

Mutiply 4.5 x 2.75 and you'll get your answer.

The temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

To get the temperature of a star with a peak wavelength of 670 nm, you can use Wien's Law, which states: T = b / λ_max where T is the temperature of the star, b is Wien's constant (approximately 2.898 x 10^6 nm K), and λ_max is the peak wavelength.
In this case, the peak wavelength (λ_max) is 670 nm. To calculate the temperature (T) of the star, follow these steps:
Step:1. Plug in the values into Wien's Law equation: T = (2.898 x 10^6 nm K) / (670 nm)
Step:2. Divide the constant by the peak wavelength: T ≈ (2.898 x 10^6) / 670
Step:3. Perform the calculation: T ≈ 4325.37 K
So, the temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

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The temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

To get the temperature of a star with a peak wavelength of 670 nm, you can use Wien's Law, which states: T = b / λ_max where T is the temperature of the star, b is Wien's constant (approximately 2.898 x 10^6 nm K), and λ_max is the peak wavelength.
In this case, the peak wavelength (λ_max) is 670 nm. To calculate the temperature (T) of the star, follow these steps:
Step:1. Plug in the values into Wien's Law equation: T = (2.898 x 10^6 nm K) / (670 nm)
Step:2. Divide the constant by the peak wavelength: T ≈ (2.898 x 10^6) / 670
Step:3. Perform the calculation: T ≈ 4325.37 K
So, the temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

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The amount that should be charged based on the demand function will be 144.5

Based on the information given, the total demand quantity is between 0 and 85, arıd demand can never be negative.

Supply: Supply se always between 0 and 100. He can create and supply 100 product quantities per month.

2. (1) Supply function: charge · 0.01 × (15²) + 0.5 × 15 = 9.75

Demand function: charge = 0.02 » (15 – 100)² = 144.5

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Specifically, we can consider the limit of the ratio:

To determine whether the sequence [tex]$a_n = \left( \frac{n}{n-6} \right)^{7n}$[/tex] converges or not, we can use the following steps:

Firstly, we can take the natural logarithm of both sides of [tex]$a_n$[/tex] to simplify the expression. Using the property [tex]$\ln(x^y) = y \ln(x)$[/tex], we have:

[tex]$$\ln \left(a_n\right)=7 n \ln \left(\frac{n}{n-6}\right)$$[/tex]

Next, we can use algebraic manipulation to rewrite the expression inside the logarithm.

Starting with the definition of the logarithm,

[tex]$$\ln \left(\frac{n}{n-6}\right)=\ln (n)-\ln (n-6)$$[/tex]

Using this identity, we can rewrite [tex]$\$ \backslash \ln \left(a_{-} n\right) \$$[/tex] as:

[tex]$$\ln \left(a_n\right)=7 n \ln (n)-7 n \ln (n-6)$$[/tex]

Now, we can use the limit comparison test to determine whether [tex]$\ln(a_n)$[/tex] converges or diverges. Specifically, we will compare [tex]$\ln(a_n)$[/tex] to a multiple of [tex]$\ln(n)$[/tex] as [tex]$n$[/tex] approaches infinity.

We can use L'Hopital's rule to find the limit of the ratio:

[tex]$$\lim _{n \rightarrow \infty} \frac{\ln (n-6)}{\ln (n)}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n-6}}{\frac{1}{n}}=\lim _{n \rightarrow \infty} \frac{n}{n-6}=1$$[/tex]

Since this limit exists and is nonzero, we can conclude that [tex]$\$ \backslash \ln \left(a_{-} n\right) \$$[/tex] and [tex]$\$ \backslash \ln (n) \$$[/tex] have the same behavior as [tex]$\$ n \$$[/tex] approaches infinity. Therefore, we can use the limit comparison test with [tex]$\$ b_{-} n=\backslash \ln (n) \$$[/tex], which we know diverges to infinity as [tex]$\$ n \$$[/tex] approaches infinity.

Specifically, we can consider the limit of the ratio:

[tex]$$\lim _{n \rightarrow \infty} \frac{\ln \left(a_n\right)}{\ln (n)}=\lim _{n \rightarrow \infty} \frac{7 n \ln (n)-7 n \ln (n-6)}{\ln (n)}$$[/tex]

Using L'Hopital's rule again, we can simplify this limit as:

[tex]$$\lim _{n \rightarrow \infty} \frac{7 n}{n} \cdot \frac{\ln (n)}{\ln (n)}-\frac{7 n}{n-6} \cdot \frac{\ln (n-6)}{\ln (n)}=7-\lim _{n \rightarrow \infty} \frac{7 n}{n-6} \cdot \frac{\ln (n-6)}{\ln (n)}$$[/tex]

We already know from our previous calculation that the limit of the fraction [tex]$\$ \backslash f r a c\{\backslash \ln (n-6)\}$[/tex] [tex]$\{\ln (\mathrm{n})\} \$$[/tex] is 1 as [tex]$\$ n \$$[/tex] approaches infinity. Therefore, the entire limit can be simplified as:

[tex]$$\lim _{n \rightarrow \infty} \frac{7 n}{n-6}=7$$[/tex]

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