Use Strong Induction to prove that: If p + 1/p E N, then Pn+1/Pn EN for all nEN.
Answers
We prove with the help of Strong Induction.
Let P(n) be the statement that Pn+1/Pn E N.
In order to prove this statement, we will utilize strong induction.So we are given that p + 1/p E N. We will show that P(n) is true for all n >= 1.
Let's consider the base case
P(1):P2/P1 = (p + 1/p)^2 - 2 = (p^2 + 2 + 1/p^2) - 2p/p = (p^2 + 1/p^2) - (2p - 2/p)
Since p + 1/p E N, both p and 1/p must be integers.
Hence, p^2 and 1/p^2 are also integers. This implies that (p^2 + 1/p^2) is an integer.
It only remains to show that (2p - 2/p) is an integer. This is equivalent to showing that 2p^2 - 2 E 0 mod p. But this is clearly true, since 2p^2 - 2 = 2(p^2 - 1) and p^2 - 1 is divisible by p.
Let's assume that P(k) is true for all k such that 1 <= k <= n. We need to prove that P(n+1) is true as well.
Now we need to prove that P(n+1) is true. In other words, we need to show that P(n+2)/P(n+1) E N, assuming that P(n+1)/P(n) E N and P(n)/P(n-1) E N.
Using the definition of P(n), we have:P(n+1)/P(n) E N and P(n)/P(n-1) E N imply that P(n+1) = aP(n) and P(n) = bP(n-1) for some integers a and b. Then:P(n+2)/P(n+1) = (P(n+2)/P(n+1)) * (P(n)/P(n)) = (P(n+2)P(n))/(P(n+1)P(n)) = (aP(n+1)P(n))/(bP(n)P(n+1)) = a/bIf we can show that a/b E N, then P(n+2)/P(n+1) E N, and P(n+1) satisfies the inductive hypothesis.
But this follows from the fact that a and b are integers and the product of two integers is always an integer.
Hence, P(n+1) is true for all n >= 1, by strong induction.Therefore, by strong induction, we have proved that if p + 1/p E N, then Pn+1/Pn EN for all nEN.
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Related Questions
∆ABC is isosceles. If m∠B = 35° what is the measure of ∠C?
Answers
Answer:
If ABC is isosceles, then <C is 35, and <A should be 110.
Step-by-step explanation:
That walks two blocks in 15 minutes how many blocks with beth walk if you worked at the same rate for an hour
Answers
According to the given statement Beth would walk approximately 8 blocks if she worked at the same rate for an hour.
To find out how many blocks Beth would walk if she worked at the same rate for an hour, we need to determine the number of blocks she walks in one minute.
Given that the person walks two blocks in 15 minutes, we can calculate the number of blocks walked in one minute by dividing 2 by 15:
2 blocks / 15 minutes = 0.1333 blocks/minute
Now, we can find out how many blocks Beth would walk in an hour by multiplying the blocks walked in one minute by 60 (the number of minutes in an hour):
0.1333 blocks/minute * 60 minutes/hour = 7.998 blocks/hour
Therefore, Beth would walk approximately 8 blocks if she worked at the same rate for an hour.
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-2x + y = 0
5x + 3y -11
Answers
Answer:
x=1, y=2
Step-by-step explanation:
if thats not the answer you need ill be happy to help again
In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. In the box, 8% contain both chocolate and nuts. Sean is allergic to both chocolate and nuts.Let C = be the event that the cookie contains chocolate.Let N = the event that the cookie contains nuts.i. Draw a Venn diagram for the given data.ii. Find the probability that a cookie contains chocolate or nuts.iii. Find the probability that a cookie does not contain chocolate or nuts.
Answers
The probability that a cookie contains chocolate or nuts is 40% and does not contain chocolate or nuts is 60%.
i. Venn Diagram:
___________________
| |
| C ∩ N |
|_______|___________|
| | |
| C | N |
|_______|___________|
ii. To find the probability that a cookie contains chocolate or nuts (denoted by C ∪ N), we can use the inclusion-exclusion principle:
P(C ∪ N) = P(C) + P(N) - P(C ∩ N)
Given:
P(C) = 36%
P(N) = 12%
P(C ∩ N) = 8%
Substituting the values:
P(C ∪ N) = 36% + 12% - 8%
= 40%
Therefore, the probability that a cookie contains chocolate or nuts is 40%.
iii. To find the probability that a cookie does not contain chocolate or nuts (denoted by the complement of C ∪ N), we can subtract the probability of C ∪ N from 100%:
P(not C ∪ N) = 100% - P(C ∪ N)
Substituting the value of P(C ∪ N) calculated in part ii:
P(not C ∪ N) = 100% - 40%
= 60%
Therefore, the probability that a cookie does not contain chocolate or nuts is 60%
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Eight identical lights are connected in series across a 110-V line. What is the voltage across each bulb? If the current is 0.56A , what is the resistance of each bulb? If the current is 0.56A , what is the power dissipated in each?
Answers
The resistance of each bulb that is connected in series with a 110 V line is 24.55 Ω with a 13.75 V Voltage difference across each bulb. The power dissipated in each bulb is 7.7 W, if 0.56 A current flows through them.
The voltage across 8 identical bulbs is 110 V
Since they are connected in a series and identical, the Voltage difference across each bulb is \(\frac{110}{8}\) V which is 13.75 V
According to the question,
The current flowing through the bulb = 0.56 A
So according to Ohm's Law,
V = IR
13.75 = 0.56 * R
R = 24.55 Ω
The power dissipated in each bulb can be calculated as
P = VI
P = 13.75 * 0.56 = 7.7 W
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In summary, the voltage across each bulb is 13.75V, the resistance of each bulb is 24.55Ω, and the power dissipated in each bulb is 7.7W.
When eight identical lights are connected in series across a 110-V line, the voltage across each bulb will be 110/8 = 13.75 V.
To find the resistance of each bulb, we can use Ohm's law, which states that resistance is equal to voltage divided by current (R = V/I). Therefore, the resistance of each bulb is 13.75/0.56 = 24.55 ohms.
To find the power dissipated in each bulb, we can use the formula P = VI, where P is power, V is voltage, and I is current. Therefore, the power dissipated in each bulb is 13.75 x 0.56 = 7.7 watts.
To find the voltage across each bulb, divide the total voltage by the number of bulbs:
Voltage across each bulb = Total Voltage / Number of Bulbs = 110V / 8 = 13.75V per bulb
Next, use Ohm's Law to find the resistance of each bulb:
Resistance (R) = Voltage (V) / Current (I) = 13.75V / 0.56A = 24.55Ω per bulb
Finally, find the power dissipated in each bulb using the formula:
Power (P) = Voltage (V) x Current (I) = 13.75V x 0.56A = 7.7W per bulb
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Evaluate: 2|x-4|+6 for x=2
Answers
\(10\)
Step-by-step explanation:\(2|x -4| +6 \text{ for } x = 2 \\ 2|2 -4| +6 \\ 2|-2| +6 \\ 2(2) +6 \\ 4 +6 \\ 10\)
Identify the sampling technique used to obtain the following sample. the first 35 students leaving the library are asked how much money they spent on textbooks for the semester. Choose the correct sampling technique below. A. Systematic sampling B. Convenience sampling C. Cluster sampling D. Stratified sampling E. Random sampling
Answers
The sampling technique used to obtain the described sample is A. Systematic sampling.
In systematic sampling, the elements of the population are ordered in some way, and then a starting point is randomly selected. From that point, every nth element is selected to be part of the sample.
In the given scenario, the first 35 students leaving the library were selected. This suggests that the students were ordered in some manner, and a systematic approach was used to select every nth student. Therefore, the sampling technique used is systematic sampling.
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2. Megan's aquarium measures 20 inches long, 14 inches wide, and 18 inches high. How many cubic inches of water would it take to completely fill the aquarium?
Answers
It would take 5040 cubic inches of water to completely fill the aquarium.
We know that the formula for the volume of cuboid :
V = length × width × height
Let us assume that l represents the length of the aquarium, w represent the width and h represents the height.
Here, l = 20 inches
w = 14 inches
and h = 18 inches
Using the formula for the volume of cuboid, the volume of aquarium would be,
V = l × w × h
V = 20 × 14 × 18
V = 5040 cu.in.
Therefore, it would take 5040 cu.in. of water.
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The area of a triangle 45.4 inches the base of the rectangle is 7 inches what is the height of the rectangle in inches ?
Answers
Answer:
r12 it has to be
Step-by-step explanation:
must be 12
\(\sqrt{8}+\sqrt{18}-\sqrt{32}\)
Answers
√8+√18−√32
√2^2·2+√3^2·2−√2^4·2
√2^ 2·√2+√3^2·√2−√2^4·√2
√2 is the answer
Graph by using the slope and the y intercept.
Answers
1. a) equation of the line :
\(y = \dfrac{2}{5}x - 7\)y - intercept = -7
so, it will pass through point (0, -7)
and if we plug the value of x as 5, we get
\(y = (\dfrac{2}{5} \times 5) - 7\)\(y = 2 - 7\)\( = - 5\)so, it will pass through point (5, -5) too
now, just plot the points (0 , -7) and (5 , -5) and join them.
2. b) equation of line is :
\(y = - 3x + 5\)here, y - intercept = 5
so the line passes through point (0 , 5)
now, Plugging the value of x = 1 we get :
\(y =( - 3 \times 1) + 5\)\(y = - 3 + 5\)\(y = 2\)so, the given line passes through point (1 , 2)
plotting the points, we can get our required line.
Which value would change the most if the number 41 were added to the set?
Answers
Answer:
Mean
Step-by-step explanation:
Our set of numbers: 50, 47, 85, 98, 35, 35, 33, 31, 27, 29
Order: 27, 29, 31, 33, 35, 35, 47, 50, 85, 98
Mean: (27+ 29 + 31 + 33 + 35 + 35 + 47 + 50 + 85 + 98) / 10 = 47
Median: (35 + 35) / 2 = 35
Mode: 35
If the number 41 were added to the set
Order: 27, 29, 31, 33, 35, 35, 41, 47, 50, 85, 98
Mean: 511 / 11 = 46.45
Median: 35
Mode: 35
So, the mean will be the value change the most if the number 41 were added to the set.
Given f(x)=x*-x³-6x², for what values of x will f(x) > 0?
Answers
The values of x will f(x) > 0 for x < 0, and f(x) < 0 for -6 < x < 0 and x > -6.
To determine the values of x for which f(x) > 0, we need to find the intervals where the function is positive. Let's analyze the function f(x) = x*-x³-6x².
First, let's factor out an x from the expression to simplify it: f(x) = x(-x² - 6x).
Now, we can observe that if x = 0, the entire expression becomes 0, so f(x) = 0.
Next, we analyze the signs of the factors:
1. For x < 0, both x and (-x² - 6x) are negative, resulting in a positive product. Hence, f(x) > 0 in this range.
2. For -6 < x < 0, x is negative, but (-x² - 6x) is positive, resulting in a negative product. Therefore, f(x) < 0 in this range.
3. For x > -6, both x and (-x² - 6x) are positive, resulting in a negative product. Thus, f(x) < 0 in this range.
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4y + 5 + 2 + 7y answer?
Answers
Answer:
11y + 7
Step-by-step explanation:
4y + 5 + 2 + 7y
4y + 7y + 5 + 2 <== combine the like terms
11y + 7
Hope this helps!
Answer:
11y+7
Step-by-step explanation:
4y+5+2+7y
combine like terms
5+2=7
then you have
4y+7+7y
combine the y's
4y+7y=11y
then you got
11y+7
The equation of a straight line is 2y=5x-2
Find the gradient and co-ordinates of the y-intercept
Answers
Answer:
2y=5x-2
Write the equation in the form of y = mx + b
where m is the gradient and b is the y-intercept.
Therefore, 2y=5x-2 will be:
y = 5/2x - 1
Gradient is 5/2
and y-intercept is (0,-1)
Which function has a greater initial value.
Answers
Answer:
Function A
Step-by-step explanation:
The Y-Intercept for Function A is 4, the initial value, where the line starts.
For Function B, there's no way to tell the Y-Intercept because there isn't a Y which is 0, but it definitely can't be bigger than 4.
A monarch butterfly averages about 50 miles per day on its migration to Mexico.
a. Complete the ratio table to show rates that are equivalent to 50 miles per day.
Time (days)
1
1
N
4
8
Distance (miles) 50
Answers
Answer:
1,50 2,100 4,150 8,400
Step-by-step explanation:
multiply each number by the time (days)
Answer:1,50 2,100 4,200 8,400
Step-by-step explanation:
If 1=50 then 2=100 just repeat that easy
Which sentence explains the correct first step in the solution of this equation?
4(x−3)=9
Answers
Step-by-step explanation:
4(x-3) = 9 EXPAND L side ( this is one way)
4x - 12 = 9
You COULD do it this way
4(x-3) = 9 DIVIDE both sides by 4
x-3 = 9/4
BUT I believe they want the first answer above ....
consider a bank which has three tellers operating in parallel, as depicted below. each of the three tellers takes a different amount of time to process each customer - teller a takes 5 minutes, b takes 8 minutes, and teller c takes 10 minutes. assume each teller handles 1 customer at a time. what is the throughput time for customers being routed to teller c (in minutes)? pick the closest answer.
Answers
The throughput time for customers being routed to teller c is 10.33 minutes so the closest answer is 10 minutes.
To calculate the throughput time for customers being routed to teller c, we need to consider the processing times of all three tellers and the arrival rate of customers.
Since the tellers are operating in parallel, the overall processing time will be determined by the slowest teller, which in this case is teller c with a processing time of 10 minutes.
Assuming a steady state where the arrival rate of customers is equal to the departure rate, the throughput time will be the sum of the processing time and the waiting time in the queue.
We can make an estimate based on the assumption that customers arrive at the bank at a constant rate. In this case, the average waiting time in the queue can be calculated using Little's Law, which states that the average number of customers in a stable system is equal to the arrival rate multiplied by the average time spent in the system.
Since there are three tellers, the arrival rate to each teller will be one-third of the overall arrival rate. Assuming an arrival rate of 1 customer per minute, the average time spent in the system for a customer routed to teller c will be:
processing time + average waiting time in queue = 10 minutes + (average number of customers in system / arrival rate) = 10 minutes + (1/3) / (1/3) = 10 minutes + 1/3 minutes = 10.33 minutes
Therefore, the closest answer to throughout time is 10 minutes.
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-a2 - 2bc -lcl if a = -3, h = -5,
and c = 2
Answers
Answer:
24
plss mark me brainliesttt
Step-by-step explanation:
6+20-2
26-2
24
Model to find -3 - -5 show eork
Answers
The mean mass of five men is 76 kg. The masses of four of the men are 72 kg, 74 kg, 75 kg and 81 kg. What is the mass of the fifth man
Answers
Answer:
78kg
Step-by-step explanation:
76×5= 380kg
380-72-74-75-81= 78kg
x=(76.5)-302
x=380-302=78
Describe the long run behavior of f(x)=5(2)x+1:
As x→−[infinity], f(x) =
As x→[infinity], f(x) =
Answers
The long run behavior of the function f(x)=5(2)x+1 is that it approaches 1 as x approaches negative infinity and it approaches infinity as x approaches positive infinity.
The long-term behavior of the function f(x)=5(2)x+1 can be discovered by examining how the function behaves as x gets closer to negative and positive infinity.
As x→−[infinity], f(x) = 5(2)^ -∞+1 = 5(0)+1 = 1
As x approaches negative infinity, the value of the function approaches 1.
As x→[infinity], f(x) = 5(2)^ ∞+1 = 5(∞)+1 = ∞
As x approaches positive infinity, the value of the function approaches infinity.
As a result, the function f(x)=5(2)x+1 behaves in the long run in such a way that it approaches 1 as x approaches negative infinity and infinity as x approaches positive infinity.
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In △ABD, BG=30 in.. What is the length of BE⎯⎯⎯⎯⎯? Enter your answer in the box. in. Triangle A B D with point G inside the triangle. A line segment begins at point B, intersects point G, and intersects the exact middle of side A D at point E. A second line segment begins at point A, intersects point G, and intersects the exact middle of side B D at point C. A third line segment begins at point D, intersects point G, and intersects the exact middle of side A B at point F.
Answers
Answer:
45cm
Step-by-step explanation:
Answer:
45
explain answer here:
well i just took the test..... love ya !!
hope u get a good grade !!
Factor out the coefficient of the variable 2d+6
Answers
Answer:
2(d+3)
Step-by-step explanation:
2*d=2d
2*3=6
Divide the following polynomial using synthetic division, then place the answer in the proper location on the grid. Write answer in descending powers of x. (x 4 - 81) (x - 3).
Answers
The final polynomial equation after synthetic division is \(x^5-3x^4-81x+243\)
We need to solve;
Put the solution in the appropriate spot on the grid after using synthetic division to divide the following polynomial. Respond with descending powers of x.
The polynomial equation given is;
\(\left(x^4-81\right)\left(x-3\right)\)
By using synthetic division;
⇒ \(x^4x+x^4\left(-3\right)-81x-81\left(-3\right)\)
\(x^5-3x^4-81x+243\)
→ The required final equation in descending powers of x is \(x^5-3x^4-81x+243\) .
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What is the ratio of roses to raises?What is the ratio of daisies to all followers in vases
Answers
The total number of flowers in the vase is 13, 4 of these are roses and the rest are daisies, then we can calculate the number of daisies by subtracting 4 from 13, then we get:
Daisises = 13 - 4 = 9
There are 9 daisies and 4 roses in the base, then the ratio of roses to daises can be written like this:
4 : 9
There are 9 daisies from a total of 13 flowers, then we can write the ratio of daisies to flowers to get:
9 : 13
Need help with this!
Answers
Answer:
488
Step-by-step explanation:
Calculate 20% of their take-home pay
3600 * 0.20 = 720
They already pay 122 and 110 per month
720 - (122 + 110)
720 - 232
488
So the largest monthly car payment they can afford is 488
Solve $\frac{1}{3} t - 5 < t - 2 \le -3t + 7.$ Give your answer as an interval.
Answers
Answer:
t ∈ (-4.5, 2.25]
Step-by-step explanation:
You want to solve the compound inequality ...
1/3t -5 < t -2 ≤ -3t +7
SplitWe can divide this into two inequalities.
1/3t -5 < t -2 ⇒ -3 < 2/3t ⇒ -4.5 < t
and
t -2 ≤ -3t +7 ⇒ 4t ≤ 9 ⇒ t ≤ 2.25
Solution intervalThe solution interval is the intersection of these two solutions:
-4.5 < t ≤ 2.25
t ∈ (-4.5, 2.25]
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Show that the 9x^2-24xy+16y^2-12x+16y-12=0 represents a pair of parallel lines .Find the distance between the lines.
Answers
First of all u will 9x^2-24xy+16y^2-12×16y-12=0 u will add the 9x^ then u will subtract 2-24