Hi there.
Soon I have my final exam in Discrete Mathematics, and I'm just working a little bit on proof by induction and refreshing my algebra. There are two exercises that I have some troubles with, and hope that you guys/girls could help me. By the way, sorry if my english is a bit rusty (norwegian).
Exercise 1:
For each natural number n, let.Prove thatfor all .
Incomplete answer (Exercise 1):
I'm going to prove that.Proof:What now? How can I make ? Maybe I have done something wrong in the process, or is it my algebra skills?
Exercise 2:
Prove by induction thatfor all . Hint: Use that .
Very incomplete answer (Exercise 2):
Base step:.The claim is true for n = 1.
Inductive step:
...
Hehe, I'm stuck right here... Hope someone can help me =D
Thanks!
One (two?) more problem(s):
Exercise 3:
I think I have solved this one, but I want a confirmation that it is correct.
Prove that is an integer for .
Answer (Exercise 3):Exercise 4:
- Let .
- , the claim is true for .
- I assume that the claim is true for some arbitrary .
- The formula is true if is a factor of 5, hence when is an integer (this is the inductive hypothesis).
- .
- is a factor of 5, therefor the claim is proven (I hope...).
.
Prove by induction that .
I understand how I can prove this, but I dont see how I can use algebra-manipulation to do this. Hope someone can help me!
Perrrrrfect
Exercise 4:
.
Prove by induction that .
I understand how I can prove this, but I dont see how I can use algebra-manipulation to do this. Hope someone can help me!
Induction way:
t(2) is evident. Assume the result for n - 1 and prove for n.
That is assume and prove
Now
Algebraic Way:
Thanks I should have understand it in the first place
Anyway, my teacher (professor) said that one task at the exam could be to prove a linear recurrence equation (at(n) + bt(n - 1) + ct(n - 2) = 0). There are no examples in the book, and he have never showed it to us. Could some one give me a task that I can solve? I know how to solve the equation, but I have never proved it...