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Questions and Answers
What is the magnitude of the force applied on the body?
What is the magnitude of the force applied on the body?
- 10N
- 0N
- 2N
- 5N (correct)
What is the displacement of the body?
What is the displacement of the body?
- 0m
- 2m (correct)
- 10m
- 5m
The work done on the body is zero.
The work done on the body is zero.
True (A)
What is the angle between the force and displacement?
What is the angle between the force and displacement?
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Prove for n = 1: What does the equation 3 + 3 + 3 + 3n equal?
Prove for n = 1: What does the equation 3 + 3 + 3 + 3n equal?
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The statement 31 = (31-1) is true.
The statement 31 = (31-1) is true.
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Assume it is true for n = k: What does the equation 3 + 3 + 3 + 3k equal?
Assume it is true for n = k: What does the equation 3 + 3 + 3 + 3k equal?
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What is the next step in proving the equation for n = k + 1?
What is the next step in proving the equation for n = k + 1?
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The equation 3 + 3 + 3 + 3k + 1 can be transformed into ___(3k - 1) + (3k + 1 - 3k - 1).
The equation 3 + 3 + 3 + 3k + 1 can be transformed into ___(3k - 1) + (3k + 1 - 3k - 1).
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Study Notes
Mathematical Induction
- The goal is to prove that
3 + 3 + 3 + 3n = 3n - 1for all positive integers n.
Base Case (n = 1)
- The equation becomes
3 + 3 + 3 + 3(1) = 3(1) - 1 - Simplifying the equation, we get
3 + 3 + 3 + 3 = 3which is true
Inductive Step (Assume true for n = k)
- We assume that the equation is true for some integer k, i.e.,
3 + 3 + 3 + 3k = 3k - 1
Prove it is true for n = k+1
- Start with the left-hand side of the equation:
3 + 3 + 3 + 3k + 3 - Rewrite it as
3(3k - 1) + (3k + 1 - 3k - 1) - Simplify to get
9/4(3k - 1) - 3/2 + 3 - Further simplify to get
9/4(3k - 1) + 3/4 + 9/4(3k - 1) - Combine like terms to get
9/4(3k) = 3/2(81k + 1) = (3k + 1) - 1 - This shows that the equation is true for
n = k + 1
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