Proving a property of a definition

I haven't felt like the professor has introduced the topic very well. I feel like I am expected to know how this works immediately but I have never done it before. I am missing the intuitive element I guess to this but moreover, I feel like sometimes the answers are too easy for me to see.

Here is first problem I worked on ... going back to the comment of the answers being overly simplified:

Fibonacci sequence

F(1) = 1

F(2) = 1

F(n) = F(n-2) + F(n-1) for n >= 3

Prove the given property of the Fibonacci sequence numbers directly from the definition.

F(n+1) + F(n-2) = 2F(n) for n >= 3

F(n+1) + F(n-2) = 2F(n)

= [F(n-2) + F(n-1)] + F(n)

= F(n) + F(n)

= 2F(n)

DONE .... this is the answer in the back of the book

All of this makes sense except for one thing ... how in the world does this demonstrate that a property works. It's easy to see that the RHS can be manipulated back and forth a bit ... it does not show anything about the LHS of the equation. Furthermore, it doesn't demonstrate that both sides are equal, which, to me, would prove its validity. What am I missing here?

Second problem that I am stuck on from the get go:

F(n) = 5F(n-4) + 3F(n-5) for n>= 6

I can't recognize a single property that I can transform this equation to ... I know very little about the fibonacci sequence other than the actual numbers so I am clueless I guess.

Thanks for any input in advance.