Questions and Answers
What is the magnitude of the force applied on the body?
What is the displacement of the body?
The work done on the body is zero.
True
What is the angle between the force and displacement?
Signup and view all the answers
Prove for n = 1: What does the equation 3 + 3 + 3 + 3n equal?
Signup and view all the answers
The statement 31 = (31-1) is true.
Signup and view all the answers
Assume it is true for n = k: What does the equation 3 + 3 + 3 + 3k equal?
Signup and view all the answers
What is the next step in proving the equation for n = k + 1?
Signup and view all the answers
The equation 3 + 3 + 3 + 3k + 1 can be transformed into ___(3k - 1) + (3k + 1 - 3k - 1).
Signup and view all the answers
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
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
A body is subjected to a force of 5N, resulting in a displacement of 2m. However, the work done is zero. What is the angle between the force and displacement?
