Fibonacci sequence problem (simple?)

[mihajovics]

Homework Statement

Write the following expression in a simpler form:
$$\sum_{1}^{n} F_{2i} \cdot F_{2i-1}$$

It doesn't have to be closed-form, probably something on the line of:
$$\sum_{0}^{n} F_{i}^{2} = F_{n} \cdot F_{n+1}$$

(We define the sequence the ususal way, starting the indexing from 0 with the first element itself being 0. So the first 6 elements from i=0 to i=5 are: 0, 1, 1, 2, 3, 5)

Homework Equations

The second equation from part 1.

The Attempt at a Solution

This is my first serious self-study effort, aimed at properly learning analysis. This problem appears in the first, introductory chapter of my book with topics like methods of proof, induction, sets. I solved all the other problems and went on to the next chapter, but this is still bugging me... :) Especially, since it's only in the introductory chapter, it's supposed to be "easy"... :)
I applied the formula in section 2. and now I'm stuck.

 

[SammyS]

What identities do you know for Fibonacci numbers?

Since F_2i = F_2i-1 + F_2i-2, you might try solving this for F_2i-1, then plug this back into your sum to get one sum minus another.

F_2i · F_2i-1 = F_2i · (F_2i - F_2i-2) = F_2i · (F_2i - F_2(i-1))

 

[mihajovics]

Thx, I'll try. For example:

Since:
$$F_{n-1} = F_{n} + F_{n-2}$$
Then:
$$F_{2i} \cdot F_{2i-1} = F_{2i}^2 - F_{2i} \cdot F_{2i-2}$$
Now use:
$$F_{n+1} \cdot F_{n-1} = F_{n}^2 + (-1)^n$$
...

We'll see :)

EDIT: why isn't the latex code showing properly? :(

 

[SammyS]

...

EDIT: why isn't the latex code showing properly? :(

There is a quirky feature to using LaTeX on this site. After hitting "Preview Post" or "Submit Reply", you must hit your browser's "Refresh button". Apparently, some cache memory doesn't get cleared properly.