we have proven that 89 divides (5³ⁿ - 6²ⁿ) for all integer n ≥ 0.
To prove that 89 divides (5³ⁿ - 6²ⁿ) for all integers n ≥ 0 using mathematical induction, we need to show that the statement holds for the base case and then demonstrate that if it holds for an arbitrary value of 'n', it also holds for 'n + 1'.
Base Case (n = 0):
Let's consider the base case where 'n = 0'. We need to show that 89 divides (5³⁽⁰⁾ - 6²⁽⁰⁾), which simplifies to 89 divides (1 - 1).
Since 89 is a factor of 0, the base case is satisfied.
Inductive Step:\
Assuming that the given statement holds for 'n = k', let's prove that it holds for 'n = k + 1'.
We assume that 89 divides [tex](5^{3k} - 6^{2k})[/tex] and want to prove that 89 divides [tex](5^{3(k+1)} - 6^{2(k+1)})[/tex].
Starting with the expression to prove:
[tex](5^{3(k+1)} - 6^{2(k+1)})[/tex]
We can rewrite this expression using the properties of exponents:
[tex](5^3 * 5^{3k}) - (6^2 * 6^{2k})[/tex]
Simplifying further:
[tex](125 * 5^{3k}) - (36 * 6^{2k})[/tex]
Now, let's use the assumption that 89 divides [tex](5^{3k} - 6^{2k})[/tex]:
Let's say [tex](5^{3k} - 6^{2k})[/tex] = 89m, where m is an integer.
Substituting this into our expression:
[tex](125 * 5^{3k}) - (36 * 6^{2k})[/tex] = (125 * 89m) - (36 * 89m)
Using the distributive property:
(125 * 89m) - (36 * 89m) = 89 * (125m - 36m)
Since (125m - 36m) is also an integer, let's call it 'p'. Therefore, we have:
89 * p
Thus, we have shown that 89 divides [tex](5^{3(k+1)} - 6^{2(k+1)})[/tex], which completes the inductive step.
By the principle of mathematical induction, the statement holds for all n ≥ 0. Hence, we have proven that 89 divides (5³ⁿ - 6²ⁿ) for all integer n ≥ 0.
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A blue and red dice are thrown simultaneously. Let A: the outcomes on the two dice are the same; B: the value on the red dice is greater or equal to the value on the blue dice. 4.1 Write down the sample pace for the experiment, and list events A and B in terms of set notation. 4.2 Determine P(A),P(B) and P(A|B)
P(A) = 6/36 = 1/6, P(B) = 21/36 = 7/12, P(A|B) = P(A∩B) / P(B) = 6/36 / 21/36 = 6/21 = 2/7. The outcomes in both A and B are: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}. The number of outcomes in A and B = 6.
4.1 Sample Space and Events in Set Notation:
When a blue and red die are thrown simultaneously, the sample space, denoted by S, consists of all possible outcomes. Since each die has six faces numbered 1 to 6, there are 36 possible outcomes in total.
Sample Space (S): {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Event A represents the outcomes where the values on both dice are the same:
A = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Event B represents the outcomes where the value on the red die is greater than or equal to the value on the blue die:
B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (4,4), (4,5), (4,6), (5,5), (5,6), (6,6)}
4.2 Calculating Probabilities:
P(A): To find the probability of event A, we divide the number of favorable outcomes (6) by the total number of outcomes (36).
P(A) = 6/36 = 1/6
P(B): To find the probability of event B, we divide the number of favorable outcomes (21) by the total number of outcomes (36).
P(B) = 21/36 = 7/12
P(A|B): To find the conditional probability of event A given event B, we need to find the probability of A and B occurring together and divide it by the probability of event B.
The outcomes in both A and B are: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
The number of outcomes in A and B = 6.
P(A|B) = P(A∩B) / P(B) = 6/36 / 21/36 = 6/21 = 2/7
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In recent years the interest rate on home mortgages has declined to less than 6.0 percent. However, according to a study by Federal Reserve Board the rate charge on credit card debit is more than 14 percent. Listed below is the interest rate charged on a sample of 10 credit cards. 14.6 16.7 17.4 17.0 17.8 15.4 13.1 15.8 14.3 14.5 Is it reasonable to conclude the mean rate charged is greater than 14 percent? Use the 0.01 significance level. Assume the interest rate on home mortgages is normally distributed.
We can conclude that, at the 0.01 significance level, there is sufficient evidence to support the claim that the mean rate charged on credit cards is greater than 14%.
How to calculate the valueThe test statistic is calculated as follows:
t = (x - μ) / (s / √n)
In this case, the sample mean is 15.66%, the sample standard deviation is 1.544%, and the sample size is 10. Plugging these values into the formula for the test statistic, we get:
t = (15.66 - 14) / (1.544 / √10)
= 3.4
The critical value is the value of the test statistic that separates the rejection region from the non-rejection region. The critical value for a two-tailed test with a significance level of 0.01 and 9 degrees of freedom (10 - 1 = 9) is 2.821.
Since the test statistic (3.4) is greater than the critical value (2.821), we reject the null hypothesis. This means that there is sufficient evidence to conclude that the mean rate charged is greater than 14%.
We can conclude that, at the 0.01 significance level, there is sufficient evidence to support the claim that the mean rate charged on credit cards is greater than 14%.
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whats 194 divided by 32
Give some memes please
Answer:
oh no homework not done
Step-by-step explanation:
mom:
PLEASE HELP QUESTION IN PHOTO WILL GIVE BRAINLIST!
Answer:
13x degrees
Step-by-step explanation:
brainliest????!!!
The weights of four puppies are shown in pounds
9.5
9 3/8
9.125
9 3/4
Which list shows these weights in order from greatest to least?
9 3/4 9.5 9 3/8 9.125
9.5 9 3/8 9 3/4
9.125 9.125 9 3/8 9.5 9 3/4 9 3/4 9 3/8 9.5 9.125
As part of a larger study investing attitudes towards relationships, a survey was administered to unmarried, currently married, and formerly married adults. First married adults had more positive perceptions of marriage than singles or remarried adults, F(2, 39) = 5.34, p = 042
It is clear that First-married adults had more positive perceptions of marriage than singles or remarried adults.
Attitudes towards relationships are often studied to determine how they affect people's perception of them. A survey was given to unmarried, currently married, and formerly married adults as part of a broader study of attitudes toward relationships. In this study, it was discovered that first-married adults had more positive attitudes toward marriage than single or remarried adults. The statistical values from the study are provided below:First married adults had more positive perceptions of marriage than singles or remarried adults, F(2, 39) = 5.34, p = 0.042.F stands for F-test, which is a statistical test used to compare whether the means of two or more groups differ from each other significantly. Here, the F-test indicated that there was a statistically significant difference in the attitudes of first-married adults, unmarried adults, and remarried adults towards marriage. Additionally, the p-value is 0.042, which indicates that there is a statistically significant difference between the groups' attitudes towards marriage.
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The sentence given "First married adults had more positive perceptions of marriage than singles or remarried adults, F(2, 39) = 5.34, p = 042" is a claim made in the larger study that was conducted investigating attitudes towards relationships.
The F(2, 39) = 5.34 indicates that the claim is statistically significant and the p-value is less than 0.05, which is the generally accepted level of significance, indicating that the findings are not due to chance.
The terms "part" and "positive" are related to the study but do not specifically apply to this claim. The claim made in this sentence is that first-married adults had more positive perceptions of marriage than singles or remarried adults. The F(2, 39) = 5.34 indicates that the claim is statistically significant. F-statistic is the ratio of between-group variance to within-group variance. Here, the between-group variance is the variance among the perceptions of different types of adults (i.e., first-married, singles, remarried) and the within-group variance is the variance within each group. Since the F-value is statistically significant, we can reject the null hypothesis and accept that there are differences in perceptions of different types of adults. The p-value is the probability of finding such results by chance. Here, the p-value is less than 0.05, which is the generally accepted level of significance, indicating that the findings are not due to chance.
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On a snorkeling trip, Dave dove at least 7 times as deep as Amy did. If Dave dove below 35 feet below the oceans surface, what was the deepest that Amy dove?
Answer:
The deepest that Amy dove below the ocean's surface is 5 feet.
Step-by-step explanation:
Let the deepest that Amy dove below the ocean's surface be x.
7x = 35
x = 5 ft
An operator at the top of a lighthouse sights a sailboat. The point from which the sighting is made is 25 m above sea level. The angle of depression of the sighting is 10 degrees. How far is the boat from the base of the lighthouse?
Answer:
141.78 m
Step-by-step explanation:
Let d = distance of boat from the base of the lighthouse
Tan 10 = 25/d
d tan 10 = 25
d = 25/tan 10
d = 141.78 m
factorize the equation 6x^2+13x+6
Answer:
(2x+3)(3x+2)
Step-by-step explanation:
.....
......
..............
Answer:
(2x + 3)(3x + 2)
Step-by-step explanation:
[tex]6x^2+13x+6 \\ \\ = 6x^2+9x + 4x+6 \\ \\ = 3x(2x + 3) + 2(2x + 3) \\ \\ = (2x + 3)(3x + 2)[/tex]
7 x 10 + 3 x 1 + 8 x
+ 5 x
100
1,000
Answer:
8x+5005073
Step-by-step explanation:
The two pentagons are similar. Find the missing side lengths.
X
12 ft
y
15 ft
21 ft
3
21 ft
N
18 ft
24 ft
Read the excerpt from The Fellowship of the Ring. On this occasion the presents were unusually good. The hobbit-children were so excited that for a while they almost forgot about eating. There were toys the like of which they had never seen before, all beautiful and some obviously magical. Many of them had indeed been ordered a year before, and all the way from the Mountain and from Dale, and were of real dwarf-make. Which detail in the excerpt identifies it as fantasy? Great presents are given at the party. The children almost forget to eat. Some of the toys are magical. The gifts have been ordered very early.
Answer:
c. some of the toys are magical
Step-by-step explanation:
The detail in the excerpt that identifies it as fantasy is "Some of the toys are magical." The correct option is 3.
What is The Fellowship of the Ring about?J.R.R. Tolkien's novel The Fellowship of the Ring was published in 1954. The Lord of the Rings is the first book in the epic fantasy series.
"Some of the toys are magical," says the excerpt, identifying it as fantasy. Magical toys imply the presence of magical elements in the story, which is a common feature of fantasy literature.
Other details, such as great gifts being given at a party and the children almost forgetting to eat, are not necessarily unique to fantasy and could be found in other genres.
The fact that the toys were ordered from the Mountain and Dale a year before.
Thus, the correct option is 3.
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Choose the monetary amount that is equivalent to 2 . $2.03 $0.30 $ 0.03 $ 2.30
Which of the following represents the function f(x) = 5x^(2) + 20x + 25 in vertex form. Identify the vertex.
The function f(x) = 5x^2 + 20x + 25 can be rewritten in vertex form as f(x) = 5(x + 2)^2 + 5, and its vertex is located at (-2, 5).
To rewrite the function f(x) = 5x^2 + 20x + 25 in vertex form, we complete the square. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
Let's complete the square:
f(x) = 5(x^2 + 4x) + 25
To complete the square, we need to add and subtract (4/2)^2 = 4 to the expression inside the parentheses:
f(x) = 5(x^2 + 4x + 4 - 4) + 25
Rearranging the terms:
f(x) = 5((x^2 + 4x + 4) - 4) + 25
Now we can factor the perfect square inside the parentheses:
f(x) = 5((x + 2)^2 - 4) + 25
Expanding and simplifying further:
f(x) = 5(x + 2)^2 - 20 + 25
f(x) = 5(x + 2)^2 + 5
Therefore, the function f(x) = 5x^2 + 20x + 25 in vertex form is f(x) = 5(x + 2)^2 + 5. The vertex is given by the coordinates (-2, 5).
The correct question should be :
What is the vertex form of the function f(x) = 5x^2 + 20x + 25? Identify the coordinates of the vertex.
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Help ASAP please!!!!!
51in
Step-by-step explanation:
15x5=75
8x3=24
75-25=51
Answer:
hi
Step-by-step explanation:
the area of the little rectangle:
3 in × 8 in= 24 in
tge area of the shaded ragion:
5in × 15in =75in
75in - 24 in=51 in
hope it helps
have a nice day
When you reflect a shape, you (blank) over an axis or line.
Answer:
poison
Step by Step Explanation
When you reflect a shape, you flip over an axis or line, the answer is flip.
What is geometric transformation?It is defined as the change in coordinates and the shape of the geometrical body. It is also referred to as a two-dimensional transformation. In the geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, and glide translation.
As we know the reflection will change the orientation not the shape or size after reflection we will get mirror image of the body.
When you reflect a shape, you flip over an axis or line.
Thus, when you reflect a shape, you flip over an axis or line the answer is flip.
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advance pile up trigonometry
Answer:
x = 6.9168 or x = 7
Step-by-step explanation:
Hope that helps :)
3. The experimental probability that Cindy will catch a fly ball is equal to 3. About what percent of the time will 7 Cindy catch a fly ball?
Correct question:
The experimental probability that Cindy will catch a fly ball is equal to 3/7. About what percent of the time will Cindy catch a fly ball?
Answer:
42.9%
Step-by-step explanation:
Given that:
Experimental probability of catching a fly is 3/7
This can be interpreted as : Out of 7 tries, Cindy caught a fly only 3 times
Expressing this as a percentage :
3/7 * 100%
0.4285714 * 100%
42.857%
= 42.9%
Hence, Cindy will catch a fly at about 42.9% of the time
Which sentence is TRUE ??
Answer:
the bottom choice
Write an equation of the line perpendicular to the given line that contains P.
Step-by-step explanation:
As the required line is perpendicular to y = (1/2)x + 5, product of slope of y = (1/2)x + 5 and required line should be - 1.
=> (1/2)m = - 1
=> m = - 2
Thus, required line is
=> y - (-8) = (-2)(x - 3)
=> y + 8 = -2(x - 3)
(i) Find the roots of f(x) = x3 – 15x – 4 using the cubic formula. : (ii) Find the roots using the trigonometric formula.
The roots using the trigonometric formula is -2 + √3
What is the cubic formula?The cubic formula is ax3 + bx2 + cx + d = 0. There is a wondering relation between the roots and the coefficients of a cubic polynomial.
The given function is
f(x) = x3 – 15x – 4
Using the Cardanos method we have
[tex]\sqrt[3]{2+11i} + \sqrt[3]{2-11i}[/tex]
Recall that the sum of the cubic root u of 2+11i with a cubic root u of 2-11i
Such that uv = -15/3 = 5
Now take u = 2+i and v = 2-i The indeed u³ = 2+11i, v³ = 2+11i and uv = 5
Therefore, 4(-u+v) is a root
But now take ω = -1/2 + √3/2i, Then ω² = -1/2 - √3i/2, ω = 1
and if you take u' = ωu, v' ω²v
u'' = ω²u, and v'' = ∈v
Then u' +v and u'' +v'' will be roots too
This means that -2±√3, v' + u' = -2 √3 and u'' + v'' = -2 +√3
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YALL JHITTS GO ADD MY T IK T OK
ITS >. * twoplayaany*
im following back
Answer:
Okay bestie ‼️
Step-by-step explanation:
Which system of inequalities has this graph as its solution?
y 10
-10
- म
5
10
-10
OA. y < 21 - 3
y
OB.y > 2 - 3
y> : +4
Ocy > 21 - 3
y < = + 4
What is the solution of 3x + 5 = 2x - 7?
Answer:
[tex]x=-12[/tex]
Step-by-step explanation:
[tex]3x+5=2x-7[/tex]
Subtract 2x from each side,
[tex]x+5=-7[/tex]
Subtract 5 from each side to find x,
[tex]x=-12[/tex]
HELPP PLSS AND NO BOTS I WILL REPORT In September, finches and jays make up more than 60% of the birds at the
feeder.
Answer:
hi
Step-by-step explanation:
Answer:
True
Step-by-step explanation:
it does add up to more then 60
1) a2 - 12a + 35
Help
Answer:
(a−5)(a−7)
Step-by-step explanation:
factor a2−12a+35
a2−12a+35
The middle number is -12 and the last number is 35.
Factoring means we want something like
(a+_)(a+_)
Which numbers go in the blanks?
We need two numbers that...
Add together to get -12
Multiply together to get 35
Can you think of the two numbers?
Try -5 and -7:
-5+-7 = -12
-5*-7 = 35
Fill in the blanks in
(a+_)(a+_)
with -5 and -7 to get...
(a-5)(a-7)
The hourly number of emergency telephone calls coming in to a police command and control centre has approximately a Normal distribution with mean of 130 and standard deviation of 25.
a) Assuming that calls arrive evenly throughout any hour and that one operator can deal with 24 calls in an hour, what is the probability that 6 operators will be able to deal with all the calls that arise in an hour? (30 marks)
b) Making the same assumptions as in (a), how many operators should there be to ensure that there is sufficient capacity to meet 95% of demand? (30 marks)
c) One possible scheme for increasing the efficiency of command and control centres is to combine the work of two such centres into one centre. For example, suppose a second centre has a similar workload to the one described above.
(i) Assuming that calls to the combined centre arrive evenly throughout any hour and that one operator can still deal with 24 calls in an hour, what is the probability that 12 operators will be able to deal with all the calls that arise in an hour? (20 marks)
(ii) Making the same assumptions again, how many operators should there be in the combined centre to ensure sufficient capacity to meet 95% of demand? (20 marks)
Given information: The hourly number of emergency telephone calls coming in to a police command and control center has approximately a normal distribution with a mean of 130 and standard deviation of 25. One operator can deal with 24 calls in an hour.
a) The probability that 6 operators will be able to deal with all the calls that arise in an hour is 0.7642.
b) The number of operators should be 203 to ensure that there is sufficient capacity to meet 95% of demand.
c) (i) The probability that 12 operators will be able to deal with all the calls that arise in an hour is 0.7852.
(ii) The number of operators should be 336 to ensure that there is sufficient capacity to meet 95% of demand.
a) Probability that 6 operators will be able to deal with all the calls that arise in an hour.
Mean, µ = 130, Standard Deviation, σ = 25.
Operator can deal with in an hour, n = 24.
Let X = number of emergency calls coming in an hour.
The number of emergency telephone calls coming in to a police command and control center in an hour can be assumed to be Poisson with λ = 130.
Since each operator can handle 24 calls in an hour, therefore, the number of operators required to handle all the calls can be obtained as follows: [tex]$$\frac{X}{24}$$[/tex].
This can be converted to a Standard Normal Variable Z using the formula:[tex]$$Z=\frac{(\frac{X}{24}-\mu)}{\sigma}$$[/tex].
Probability that 6 operators will be able to deal with all the calls that arise in an hour can be calculated as follows:
[tex]$$\begin{aligned} \frac{X}{24} &\leq 6 \\ X &\leq 6 \times 24 \\ X &\leq 144 \end{aligned}$$[/tex]
Now, we need to find the probability of Z ≤ [tex]$$(\frac{144}{24}-130)/25=0.72$$[/tex].
Using normal distribution tables, we get P(Z ≤ 0.72) = 0.7642.
Hence, the probability that 6 operators will be able to deal with all the calls that arise in an hour is 0.7642.
b) To find the number of operators should there be to ensure that there is sufficient capacity to meet 95% of demand.
Let X = number of emergency calls coming in an hour.
The number of emergency telephone calls coming in to a police command and control center in an hour can be assumed to be Poisson with λ = 130.
Since each operator can handle 24 calls in an hour, therefore, the number of operators required to handle all the calls can be obtained as follows:[tex]$$\frac{X}{24}$$[/tex].
This can be converted to a Standard Normal Variable Z using the formula:[tex]$$Z=\frac{(\frac{X}{24}-\mu)}{\sigma}$$[/tex].
To ensure sufficient capacity to meet 95% of demand, we need to find the value of X such that: P(X ≤ x) = 0.95.
Using the Z table, we can find that the probability of Z ≤ 1.645 is 0.95.
Now, we can use the formula:
[tex]$$\frac{X}{24}-130/25=1.645$$[/tex]
[tex]$$X= 1.645\times 25\times 24+130$$[/tex]
[tex]$$X=202.63$$[/tex]
Therefore, the number of operators should be 203 to ensure that there is sufficient capacity to meet 95% of demand.
c) Two centers are combined and let X_1 and X_2 be the number of calls at centers 1 and 2, respectively.
Then the total number of calls, X = X_1 + X_2, follows a normal distribution with
mean = 130 + 130
mean = 260, and
standard deviation = sqrt(25^2 + 25^2)
= 35.36
i) Probability that 12 operators will be able to deal with all the calls that arise in an hour can be calculated as follows:
[tex]$$\begin{aligned} \frac{X}{24} &\leq 12 \\ X &\leq 12 \times 24 \\ X &\leq 288 \end{aligned}$$[/tex]
Now, we need to find the probability of Z ≤ [tex]$$(\frac{288}{24}-260)/35.36=0.789$$[/tex].
Using normal distribution tables, we get P(Z ≤ 0.789) = 0.7852.
Hence, the probability that 12 operators will be able to deal with all the calls that arise in an hour is 0.7852.
ii) To ensure sufficient capacity to meet 95% of demand, we need to find the value of X such that: P(X ≤ x) = 0.95.
Using the Z table, we can find that the probability of Z ≤ 1.645 is 0.95.
Now, we can use the formula:
[tex]$$\frac{X}{24}-260/35.36=1.645$$[/tex]
[tex]$$X= 1.645\times 35.36\times 24+260$$[/tex]
[tex]$$X=335.58$$[/tex]
Therefore, the number of operators should be 336 to ensure that there is sufficient capacity to meet 95% of demand.
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Find mzw.
X
po
w
(x + 12°
Need help
Answer:
∠ W = 51°
Step-by-step explanation:
Since WX is the diameter , then
∠ WYX = 90° ( angle in a semicircle )
The sum of the 3 angles in a triangle = 180° , then
x + x + 12 + 90 = 180
2x + 102 = 180 ( subtract 102 from both sides )
2x = 78 ( divide both sides by 2 )
x = 39
Then
∠ W = x + 12 = 39 + 12 = 51°
A skyscraper that stands 620 feet cast of shadow that is 125 feet long. What is the ratio of the shadow to the height of the skyscraper?