given that t(n) = 1 if n=0 and t(n) = t(n-1) 2n otherwise; show, by induction, that t(n) = 2n 1 -1. show all three steps of your induction explicitly
Answers
To prove the given statement t(n) = 2^n - 1 using induction, we need to follow three steps: the base case, the induction hypothesis, and the induction step.
1. Base case (n=0):
Given that t(0) = 1. Let's check if it satisfies the formula t(n) = 2^n - 1:
t(0) = 2^0 - 1 = 1 - 1 = 0
Since t(0) = 1 and the formula gives t(0) = 0, there's a typo in the student question. The correct formula should be t(n) = 2^n - (-1), which makes t(n) = 2^n + 1.
Now, let's check the base case with the corrected formula:
t(0) = 2^0 + 1 = 1 + 1 = 2
The base case holds for the corrected formula since t(0) = 2.
2. Induction hypothesis:
Assume the statement is true for some n=k, i.e., t(k) = 2^k + 1.
3. Induction step (n=k+1):
We need to prove that the statement holds for n=k+1, i.e., t(k+1) = 2^(k+1) + 1.
Using the given recursive formula: t(n) = t(n-1) * 2n
t(k+1) = t((k+1)-1) * 2(k+1) = t(k) * 2(k+1)
Substitute the induction hypothesis into the equation:
t(k+1) = (2^k + 1) * 2(k+1)
Now let's show that this equals 2^(k+1) + 1:
t(k+1) = (2^k * 2(k+1)) + 2(k+1) = 2^(k+1) * 2 - 2 + 2 = 2^(k+1) + 1
Thus, we have proven the statement t(n) = 2^n + 1 by induction.
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Related Questions
What type of motion can be seen in a hurricane that cannot be seen in the movement of a weather front? translation periodic motion rotation
Answers
Answer: Rotation
----------------------------------------
(Apologies since this is late; hope future students find this helpful!)
Answer:
Rotation
Explanation:
Though two types of movement (rotation and translation) take place in a hurricane as it moves, this question refers to rotation.
If a star is 720 , 000 , 000 , 000 , 000 meters from Earth, how many seconds does it take light to travel from the Earth to the star
Answers
Answer:
anywhere between 100000 to about 400000 human years .
Explanation:
The high luminosity of the galactic center region suggests that ____
Answers
The high luminosity of the galactic center region suggests that there is a high concentration of stars and other celestial objects emitting light in that area.
This high luminosity can be attributed to the presence of a supermassive black hole, dense star clusters, and active star formation processes, all contributing to the increased brightness observed in the galactic center region.
According to the idea behind active galactic nuclei, heated plasma in an accretion disc that revolves around the black hole is principally responsible for the active galactic nucleus' high luminosity.
An active galactic nucleus (AGN) is a compact area at the Centre of a galaxy that exhibits features that indicate the luminosity is not coming from stars and is substantially brighter than usual over at least some of the electromagnetic spectrum. Is called active galactic nucleus.
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What water pressure must a pump that is located on the first floor supply to have water on the thirteenth of a building with a pressure of 35 lb/in2 Assume that the distance between each floors is 10ft.
Answers
The water pressure on the first floor must be 455 PSI in order to push the water to the 13th floor at the given pressure.
The given parameters;
Pressure on the 13 th floor, P₁ = 35 PSIDistance between each floor, d = 10 ftThe vertical pressure of the water is calculated as follows;
\(P = \rho gh\\\\\frac{P}{h} = \rho g\\\\\frac{P}{h} = k\\\\\frac{P_1}{h_1} = \frac{P_2}{h_2} \\\\\)
The vertical height of the first floor from the 13th floor = 130 ft
The vertical height of the 13 ft floor = 10 ft
\(P_1 = \frac{P_2 h_1}{h_2} \\\\P_1 = \frac{35 \times 130}{10} \\\\P_1 = 455 \ PSI\)
Thus, the water pressure on the first floor must be 455 PSI in order to push the water to the 13th floor at the given pressure.
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when light rays traveling in air at a specific angle interact with water, the light rays begin to slow down and bend slightly . this phenomenon is known as _______________.
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When light rays traveling in air at a specific angle interact with water, the light rays begin to slow down and bend slightly . this phenomenon is known as refraction.
Refraction is the bending of light as it passes through a medium with a different refractive index, such as from air to water. The speed of light is different in different media due to their different refractive indices, and the change in speed causes the light to change its direction of travel.
When light rays travel from air to water, the refractive index of water is higher than that of air, so the light rays slow down and bend towards the normal (the imaginary line perpendicular to the surface of the water) as they enter the water. This is why objects submerged in water appear to be in a different position than they actually are when viewed from above the surface. Refraction is an important phenomenon in optics and is used in lenses and other optical devices.
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A body weighing 188N moves with a speed of 5m/s in a horizontal circular path of radius 5m. Calculate the magnitude of the centripetal force acting on the body
Answers
The centripetal acceleration a of the body is
a = v ² / R
where
• v = tangential speed of the body
• R = radius of the circle
Substituting these into the acceleration gives
a = v ² / R = (5 m/s)² / (5 m)
a = 5 m/s²
We're given the body's weight W = 188 N, so its mass is m = W / g. Use Newton's second law to get the magnitude of the centripetal force:
F = m a = (W / g) a ≈ (188 N) / (9.80 m/s²) (5 m/s²)
F ≈ 95.9 N
primates rely primarily on _______________ to negotiate their environment.
Answers
Primates rely primarily on vision to negotiate their environment.
As highly adaptable mammals, primates possess a keen sense of sight which enables them to successfully interact with their surroundings. This exceptional visual acuity is mainly due to their well-developed eyes, which allow for excellent color discrimination, depth perception, and the ability to see in low-light conditions.
Primates possess forward-facing eyes, a feature that contributes to the enhancement of their depth perception. This is crucial for tree-dwelling species that must accurately judge distances when jumping between branches or reaching for food. Additionally, the majority of primates have trichromatic vision, meaning they can see and differentiate a wide range of colors. This ability helps them identify ripe fruits and detect potential predators or threats.
Furthermore, primates have a larger brain size in comparison to other mammals, which aids in the processing of visual information. The visual cortex, the part of the brain responsible for processing visual data, is particularly well-developed in primates, enabling them to efficiently interpret and respond to their environment.
In summary, primates rely primarily on vision to navigate and adapt to their surroundings. Their advanced visual capabilities, such as enhanced depth perception, color discrimination, and low-light vision, coupled with a well-developed visual cortex, allow them to successfully interact with their environment and thrive in a variety of habitats.
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which of the following statements about gravitational waves are true?
a. The emission of gravitational waves from merging black holes is predicted by Einstein's general theory of relativity.
b. The first direct detection of gravitational waves, announced in 2016, came from the LIGO observatory.
c. Two orbiting neutron stars or black holes will gradually spiral toward each other as a result of energy being carried away by gravitational waves.
d. The emission of gravitational waves from merging black holes is predicted by Newton's universal law of gravitation.
e. Although gravitational waves are an important theoretical prediction, we do not yet have any observational evidence that they exist.
Answers
Two orbiting neutron stars or black holes will gradually spiral toward each other as a result of energy being carried away by gravitational waves.
How do you explain gravitational waves?"Spacetime ripples were gravitational waves. When things move, the curvature of gravity alters, and these changes propagate outward as gravitational waves (like ripples on a pond). A supermassive black hole is a stretch & compress of space and hence can be determined by calculating the current in length among two objects.”
How do gravitational waves affect us?Even from the distance of the edge of the universe, cosmic rays would pass by us almost completely undetectable. Although these waves in spacetime convey more force than any other global catastrophe, the connections are so faint that they scarcely influence us.
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describe two forces you experience/observer everday
Answers
Answer:
Static friction: Keeps objects from sliding
Kinetic energy: Simply walking
Explanation:
Answer:
Friction and Inertia
Explanation:
Friction - Friction causes a force on a surface which is in the opposite direction to its motion. Example: applying the brakes in a car.
Inertia - An object will stay in motion and an object at rest will stay at rest unless acted upon by an outside force. Example: Why your book stays still on the table if you don't touch it, or why you shoot forward when brakes are applied in a car.
After the switch has been closed for a very long time, it is then opened. What is Q(topen), the charge on the capacitor at a time topen = 623 us after the switch was opened?
Answers
After the switch has been closed for a very long time, it is then opened. What is Q(open), the charge on the capacitor at a time open = 623 us after the switch was opened.
After a very long time, a capacitor is completely charged, and when the switch is turned on, the capacitor discharges in the form of an electric current, and the capacitor charges up again when the switch is turned off. Thus, the capacitor's charge Q is calculated using the formula:
Q(t) = C×V(t),
where t is the time,
C is the capacitance of the capacitor,
and V(t) is the voltage across the capacitor at time t.
In this case, the time is open = 623 us after the switch was opened.
The voltage across the capacitor at this moment, V(open), can be obtained using the formula:
\(V_{o} = V e^{-t_o/RC}\)
where V0 is the initial voltage across the capacitor, R is the resistance of the circuit, and C is the capacitance of the capacitor.
Substituting the given values,
we have \(V_o= 15 V*e^{-623*10^{-6} )/(15*10^{3} *10^{-6})\)
= 13.61 V
The charge on the capacitor at the time open can now be obtained by multiplying the capacitance of the capacitor by the voltage across it at this time:
Q(topen) = C × V(topen)
\(= 12*10^{-6} F *13.61 V\)
= 0.1633 mC (or 163.3 µC)
Thus, the charge on the capacitor at the time open = 623 us
after the switch was opened is 0.1633 mC or 163.3 µC.
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All scientific inquiry is done using the ______________ method.
Answers
Answer:
The scientific method
Make an observation.
Ask a question.
Form a hypothesis, or testable explanation.
Make a prediction based on the hypothesis.
Test the prediction.
Iterate: use the results to make new hypotheses or predictions.
Explanation:
HELP ME ASAP PLZZZ NOW!!!
Answers
Answer:
B. It's nucleus releases energy
Explanation:
A roller coaster car is released from a standing start at the top of a 20.0-m hill. The car rolls down the hill to a point 10.0 m above the ground, then up a second hill to 15.0 m, and then down again to point X at 8.00 m above the ground. Use what you know about the sum of the kinetic and potential energies to find the car's speed at X.
A.
11.7 m/s
B.
12.5 m/s
C.
15.3 m/s
D.
19.8 m/s
Answers
Answer:
21.5 i did the assesment
Explanation:
If you travel 450 meters in 40 seconds, what is your average speed in meters per
second?
A)11.25
B)40
C)450
D)18000
Answers
Answer:
The answer is a 11.25m/s
Ma of a machine is always less than Vr
Answers
Answer:
MA of a machine is always less than VR.
Explanation:
It is because mechanical advantage decreases due to the friction and weight of moving parts of the machine whereas the velocity ratio remains constant.
If an electron is released at the equator and falls toward earth under the influence of gravity, the magnetic force on the electron will be toward the
Answers
The magnetic force on the electron will be west. Example: If it is dropped above the equator in Africa, the electron will head towards North America.
Dùng công thức gia tốc trọng trường g =GM/R^2 . Tìm gia tốc trọng trường gH trên sao Hỏabiết khối lượng sao Hỏa bằng 0,5325 bán kính Trái Đất, theo đơn vị m/s^2 . Biết gia tốc trọng trường Trái Đất gTĐ =9,8 m/s^2 .
A. 1,204
B. 0,305
C. 3,712
D. 6,218
Answers
after point c, friction acts on the roller coaster and slows it to a stop. what is the work done by friction?
Answers
The roller-coaster has a speed of 47.7 m/s at point B. To bring the roller-coaster to a stop at point D, an average force of 10,784 N is required.
We can use conservation of energy to solve this problem. At Point A, the roller-coaster has gravitational potential energy, which is converted into kinetic energy as it moves down the track. At Point C, the roller-coaster has only kinetic energy, which is converted into thermal energy as it comes to a stop at Point D.
The initial potential energy at Point A is given by
PE = mgh = 850 kg * 9.81 m/s² * 140 m = 1,106,460 J
At Point B, the roller-coaster has lost some potential energy and gained an equal amount of kinetic energy. Using conservation of energy, we can find the kinetic energy and then the velocity
PE(A) = KE(B)
1,106,460 J = (1/2) * 850 kg * v²
v = √(2 * 1,106,460 J / 850 kg) = 47.7 m/s
The roller-coaster then travels along the straight track from Point C to Point D. During this time, it experiences a net force that slows it down. We can use the work-energy principle to find the average force required to bring it to a stop
Work done by brakes = change in kinetic energy
F_avg * d = (1/2) * m * v²
F_avg = (1/2) * m * v² / d
where d is the distance from Point C to Point D, which is 120 m.
The final velocity at Point C can be found by equating its potential energy at Point D to its kinetic energy
PE(D) = (1/2) * m * v_f²
1,106,460 J - (850 kg * 9.81 m/s² * 80 m) = (1/2) * 850 kg * v_f²
v_f = √(2 * (1,106,460 J - 850 kg * 9.81 m/s² * 80 m) / 850 kg) = 36.6 m/s
Substituting the values, we get
F_avg = (1/2) * 850 kg * (47.7 m/s)² / 120 m = 10,784 N (approximately)
Therefore, the roller-coaster is moving at 47.7 m/s at Point B, and an average force of 10,784 N is required to bring it to a stop at Point D.
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--The given question is incomplete, the complete question is given
" An 850 kg roller-coaster is released from rest at Point A of the track is one wave and straight path. Assume there is no friction or air resistance between Points A and C. How fast is the roller-coaster moving at Point B? What average force is required to bring the roller-coaster to a stop at Point D if the brakes are applied at Point C? A is starting point at 140 m height from horizontal base, B s the point on wave at height 95m, C is starting point of straight line in track CD is 120 m, D is the final point on height 80 m"--
An electron moves in an electric field. Does it move toward regions of higher potential or lower potential? explain.
Answers
The electric fields is always directed from higher potential to the lower potential.
An electron moves in an electric field. Does it move toward regions of higher potential or lower potential?
The electric fields always aiming from the region of higher potential to the region of lower potential. So the force of electrostatic and the direction of the travel of electrons will be always from lower potential to the region of higher potential.
An electric field is defined as the physical field which covers the electrically charged particles and also they can apply force on all other charged particles.
So we can conclude that the electric fields is always directed from higher potential to the lower potential.
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The minute hand on a watch is 3. 00 cm in length. Assume the origin of the xy -plane is at the center of the clock face, the x -axis points in the same direction as the minute hand at 8:15 a. M. , and the y -axis points in the same direction as the minute hand at 8:00 a. M
Answers
To determine the position of the minute hand at 8:15 a.m., we can use the following approach:
Convert the time to radians: At 8:15 a.m., the minute hand is 75% of the way from the 8:00 a.m. mark to the 9:00 a.m. mark. Since the minute hand travels 360 degrees in 60 minutes, it travels 6 degrees per minute. Therefore, at 8:15 a.m., it has traveled 90 degrees from the 8:00 a.m. mark. This is equivalent to (90/360) x 2π radians = π/2 radians. Use trigonometry to determine the position: Since the minute hand is 3.00 cm long and pointing in the direction of the positive x-axis at 8:15 a.m., its position can be described by the coordinates (3.00 cos(π/2), 3.00 sin(π/2)) = (0, 3.00). Therefore, the minute hand is pointing straight up at 8:15 a.m. Note that the x-axis points in the same direction as the minute hand at 8:15 a.m., while the y-axis points in the same direction as the minute hand at 8:00 a.m. This means that the x-axis is perpendicular to the minute hand at 8:00 a.m., and the y-axis is perpendicular to the minute hand at 8:15 a.m.
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A train accelerated from 20km/hr to 80km/hr in 4 minutes. How much distance does
it cover in this period? Assume that the tracks are straight?
Answers
Answer:
S = V0 t + 1/2 a t^2
a = (V2 - V1) / t = (80 - 20) km/hr / .067 hr = 900 km/hr^2
S = 20 km/hr * 067 hr + 1/2 * 900 * .067^2 km = 3.34 km
characteristics of saturn
Answers
Saturn is a planet in solar system. It's ring is made out of small covered ice particle. It has over 30 moons.
made up of hydrogen and helium
second biggest planet in the solar system
least dense planet in the solar system
This was merely an accident and I did not mean to post a question.
Answers
Answer:
ok a good rest of your day
Explanation:
+
1. Energy due to motion is
_______ energy
Answers
Answer:
Kinetic energy
Explanation:
The energy associated with an object's motion is called kinetic energy. A speeding bullet, a walking person, and electromagnetic radiation like light all have kinetic energy.
Kinetic energy. For example, a walking person.
how to find average velocity on a velocity time graph
Answers
To find the average velocity on a velocity-time graph, you need to calculate the slope of the line connecting two points on the graph. The average velocity represents the change in velocity divided by the change in time between those two points.
To calculate the average velocity, you can use the formula:
Average velocity = (change in velocity) / (change in time)
You can determine the change in velocity by finding the difference between the final velocity and the initial velocity. The change in time is the difference in the time coordinates of the two points.
Select two points on the velocity-time graph, typically denoted by (t₁, v₁) and (t₂, v₂), where t represents time and v represents velocity. Then, substitute the values into the formula mentioned above to calculate the average velocity.
It's important to note that the average velocity provides information about the overall change in velocity over a specific time interval, rather than instantaneous velocity at a particular moment.
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Seeing these new artifically designed creatures at home, as the result of breeding, he coined the term ____ ______ to explain the variation of species around the islands and (by extension of his theory) also around the world.
Answers
Answer: Artificial Selection
Explanation: Artificial Selection is described as the identification of desirable traits in both plants and animals, then enhance them, so that those traits can be manifested in future generations.
This is the scope behind cross breeding of plants and animals and this have largely been used in agricultural sector to improve input (such as disease-resistant seeds) and other biotechnological settings.
Darwin's described artificial selection as the human form of natural selection.
Describe what it means to view something from a frame of reference. Give an example to illustrate your explanation. (4 points)
Answers
An ideal battery is a source of fixed: (select all that apply) EMF Voltage Resistance Current
Answers
An ideal battery is a source of fixed EMF and voltage. An ideal battery is a theoretical construct that is used to understand how batteries work. It is a model that assumes that a battery has no internal resistance and that its voltage does not change with time.
In this idealized scenario, a battery is a source of fixed electromotive force (EMF) and voltage. The EMF is a measure of the battery's ability to produce electrical energy, while the voltage is the measure of the potential difference between the two terminals of the battery. In an ideal battery, the voltage is constant and does not depend on the amount of current that is drawn from it. This means that an ideal battery can deliver a constant amount of energy over an extended period of time.
While an ideal battery is not a real device, it is a useful tool for understanding the behavior of real batteries. In real batteries, there is always some internal resistance that causes the voltage to drop as current is drawn from the battery. This means that the actual voltage of a battery depends on the amount of current that is being drawn from it. Moreover, the voltage of a real battery decreases over time as the battery discharges. Nevertheless, the concept of an ideal battery is helpful in understanding the principles of battery operation and in designing practical battery systems that can deliver reliable and consistent performance.
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An object starts in the negative direction direction with a speed a 2m/s but is accelerating in the positive direction with acceleration of 3 m/s^2. what is the velocity and displacement at the end of 10 seconds
Answers
After 10 seconds, the object's displacement can be calculated as follows: An object has a positive velocity when it is moving in the right direction.
If an object is slowing down, its acceleration vector is pointing away from the direction in which it is moving (in this case, a negative acceleration).
Initial speed (u) equals 2 m/s
Final speed (v) is 32 m/s.
The speed of sound is 3 m/s2.
Position (s) =? v2 = u2 + 2as
32² = 2² + (2 × 3 × s)
1024 = 4 + 6s
assemble similar terms
1024 - 4 = 6s
1020 = 6s
6 divided by both sides
s = 1020 / 6
s = 170 m
The displacement is 170 m as a result.
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63 J of heat are added to a closed system. The initial energy of the system is 58J, and the final internal energy is 93J. How much work is being done by the system?
Answers
ΔU = Q + W
heat in, Q + heat out, Q - does work , W - work in, W +93-58 = 63 + W
35 = 63 + W
W = - 28 J (does work/being done by system)