Prove that recurrence relation is odd

[njama]

Homework Statement

Hello!

I just want to be sure if I did solve the task correctly.

The task is:

Prove that b_n is odd for integers n>=1

b₁=1
b₂=3
b_k=b_k-2+2b_k-1 for k>=3

Homework Equations

The Attempt at a Solution

Induction basis:

b₁=1 true 1 is odd
b₂=3 true 2 is odd

Now the question is:

Could I use the strong (complete) induction?

If I can use it, the solution is simple:

Let the recurrence relation is true for all k, such that n<k and n=k i.e n<=k
b_k=b_k-2+2b_k-1

then for n=k+1

b_k+1=b_k-1+2b_k

b_k-1 is odd and 2b_k is even therefore odd+even=odd

Is this correct?

Thank you.

Regards.

 

[susskind_leon]

I think you pretty much figured it out. Check this page if you're not quite sure what you are doing http://en.wikipedia.org/wiki/Mathematical_induction