Well, I have the following two problems involving Fibonacci sequences and Lucas numbers.
I know that they share the same technique, but I don't have clear the procedure:
$$f_n = f_{n-1} + f_{n-2}: f_0 =0, f_1=1$$
$$l_n=l_{n-1} +l_{n-2}:l_0=2,l_1=1$$
Now, I want to prove that:
$$\sum\limits_{k=0}^nf_k= f_{n+2}-1 $$
$$\sum\limits_{k=0}^n l_k^2= l_nl_{n+1} +2$$
My question is, what kind of technique should be used to deal with such problems?