I'm not sure how to go about setting up the steps for this problem.
a1 = 1, a2 = 1 and an = 2an-1 + an-2 for all n ≥ 3. prove an is odd for all n≥1.
first six are 1,1,3,7,17,41
I could also use help on another problem...
proving (3,5,7) is the only prime triple - consecutive odd integers which are all prime -
to do this i prove that for any odd integer k, one of the integers 2k+1, 2k+3, and 2k+5 must be divisible by 3.
I know i need to use cases for this.
how does this then show that (3,5,7) is the only prime triple?
thanks for the input guys.
The characteristic equation is
It has solutions
Use the initial conditions to solve:
Solve the system and get
Therefore, we have:
Now, perhaps you can use this to prove the solutions are always odd.
We have the base cases, so we'll go to the inductive step
Suppose and are odd, we shall prove that is also odd
Or if you prefer: and ( and are natural numbers)
So an odd number