Please help....

Question

If this pattern continues, fill in the blank.

 

1 to the 2nd power + 1 to the 2nd power + 2 to the 2nd power + .............+ F n to the 2nd power =

F n x ______?

 

Note:  n is below the F (can't do that on here). Does this make any sense? Thank you so much!

Answer 1

Hi Karen,

 

It looks like that you're working with here is the Fibonacci sequence, which basically looks like this:

1 + 1 + 2 + 3 + 5+ 8 + 13+ ....

Each term in the Fibonacci sequence is the sum of the two terms before it. So, F_n= F_n-1+ F_n-2

So, how is that related to this? Well, here you're adding up the squares of each Fibonacci term. Let's do a few of these terms and see if we notice a pattern:

1² + 1² + 2²=6
1² + 1² + + 2² + 3² =15
1² + 1² + + 2² + 3² + 5² =40
1² + 1₂ + + 2² + 3² + 5² + 8²= 104

What I'm going to do now is factor these terms:

6 = 3 *2
15 = 3 * 5
50 = 5 * 8
104 = 8 * 13

If you notice, we're seeing a pattern where the sum of the squares of the sequence is the product of the the nth term in the Fibonacci sequence times the the next term in the sequence. So, we can say that the sum can be written as:

F_n* F_n+1

To prove it, let's plug in what we figured out to be the sum and see if the sum would be the same for the next term- this is called a proof by induction:

(F_n* F_n+1) = our sum of the sequence up to this point

(F_n+1)² = the next term in the sequence

So, we can write the sum of the sequence when adding the next term as:

(F_n*F_n+1) + (F_n+1)²

We can factor out an F_n+1 term:

F_n+1(F_n+F_n+1) 

Based on our definition of what F_n is, we  can say that the terms in the parentheses combine to F_n+2. So, our new product is:

= F_n+1* F_n+2

 

Which would be the sum of the sequence if we added the next term, based on what we found last time. Thus you have found a way to define the sum of the squares of the Fibonacci sequence, which would be F_n+F_n+1

Hope this helps!