# Thread: Fibonacci sequence

1. ## Fibonacci sequence

Would the formulas

$F_{n+3} + F_{n} = 2F_{n+2}$ and $F_{n+3} - F_{n} = 2F_{n+1}$ for n=0,1,2,...

necessarily remain true if the sequence
Fn was replaced by a sequence with the same recurrence relation as the Fibonacci sequence but different initial terms? Justify your answer briefly.

I'm not sure, but would the formula remain true, but with different initial terms the sequence would have different numbers? I'm not quite sure how to tackle this. Thanks for any pointers.

2. I think those can be deduced from the basic recurrence relation $F_{n+2}=F_{n+1}+F_n$. If you start substituting on the left-hand side, the right-hand side comes out pretty quickly in both cases.

Post again in this thread if you're still having trouble.

3. Originally Posted by hollywood
I think those can be deduced from the basic recurrence relation $F_{n+2}=F_{n+1}+F_n$. If you start substituting on the left-hand side, the right-hand side comes out pretty quickly in both cases.

Post again in this thread if you're still having trouble.
Hi, me again! Sorry, but I am still unsure of the answer.

I know from earlier workings that $F_{n}=F_{n+2}-F_{n+1}$ and $F_{n+3}=F_{n+2}+F_{n+1}$ but I am not sure how to answer the question.

Any help is very much appreciated

4. Originally Posted by cozza
Would the formulas

$F_{n+3} + F_{n} = 2F_{n+2}$ and $F_{n+3} - F_{n} = 2F_{n+1}$ for n=0,1,2,...

necessarily remain true if the sequence
Fn was replaced by a sequence with the same recurrence relation as the Fibonacci sequence but different initial terms? Justify your answer briefly.

I'm not sure, but would the formula remain true, but with different initial terms the sequence would have different numbers? I'm not quite sure how to tackle this. Thanks for any pointers.
Originally Posted by cozza
Hi, me again! Sorry, but I am still unsure of the answer.

I know from earlier workings that $F_{n}=F_{n+2}-F_{n+1}$ and $F_{n+3}=F_{n+2}+F_{n+1}$ but I am not sure how to answer the question.

Any help is very much appreciated
These are essentially the same formula. If you add $F_{n+1}$ to both sides of the first formula, you get $F_{n}+F_{n+1}=F_{n+2}-F_{n+1}+F_{n+1}=F_{n+2}$ so $F_{n+2}=F_{n+1}+F_{n}$, which is the second formula with n replaced by by n-1 (i.e. all three subscripts decreased by 1).

But let's take your two formulas and do some algebra:

$F_{n}=F_{n+2}-F_{n+1}$

$F_{n+3}=F_{n+2}+F_{n+1}$

adding $F_{n+1}$ to the first gives:

$F_{n}+F_{n+1}=F_{n+2}$

and substituting into the second gives:

$F_{n+3}=F_{n}+F_{n+1}+F_{n+1}=F_{n}+2F_{n+1}$

and subtracting $F_{n}$ from both sides gives:

$F_{n+3}-F_{n}=2F_{n+1}$.

That's the second formula you were looking for.

Now, if you take:

$F_{n+3}=F_{n+2}+F_{n+1}$

and instead of substituting for $F_{n+2}$, you substitute for $F_{n+1}$ using $F_{n+1}=F_{n+2}-F_{n}$, you get:

$F_{n+3}=F_{n+2}+F_{n+2}-F_{n}$

$F_{n+3}+F_{n}=2F_{n+2}$, which is the first formula you were looking for.

- Hollywood

5. Originally Posted by hollywood
These are essentially the same formula. If you add $F_{n+1}$ to both sides of the first formula, you get $F_{n}+F_{n+1}=F_{n+2}-F_{n+1}+F_{n+1}=F_{n+2}$ so $F_{n+2}=F_{n+1}+F_{n}$, which is the second formula with n replaced by by n-1 (i.e. all three subscripts decreased by 1).

But let's take your two formulas and do some algebra:

$F_{n}=F_{n+2}-F_{n+1}$

$F_{n+3}=F_{n+2}+F_{n+1}$

adding $F_{n+1}$ to the first gives:

$F_{n}+F_{n+1}=F_{n+2}$

and substituting into the second gives:

$F_{n+3}=F_{n}+F_{n+1}+F_{n+1}=F_{n}+2F_{n+1}$

and subtracting $F_{n}$ from both sides gives:

$F_{n+3}-F_{n}=2F_{n+1}$.

That's the second formula you were looking for.

Now, if you take:

$F_{n+3}=F_{n+2}+F_{n+1}$

and instead of substituting for $F_{n+2}$, you substitute for $F_{n+1}$ using $F_{n+1}=F_{n+2}-F_{n}$, you get:

$F_{n+3}=F_{n+2}+F_{n+2}-F_{n}$

$F_{n+3}+F_{n}=2F_{n+2}$, which is the first formula you were looking for.

- Hollywood
So presumably the formulas would remain true if the sequence Fn was replaced by a sequence with the same recurrence relation as the Fibonacci sequence but different initial terms, and that is proved by replacing the initial term

$F_{n+2}=F_{n+1}+F_n$ with F_{n+3}+F_{n}=2F_{n+2}[/tex] and F_{n+3}-F_{n}=2F_{n+1}[/tex]

6. Originally Posted by cozza
So presumably the formulas would remain true if the sequence Fn was replaced by a sequence with the same recurrence relation as the Fibonacci sequence but different initial terms, and that is proved by replacing the initial term

$F_{n+2}=F_{n+1}+F_n$ with F_{n+3}+F_{n}=2F_{n+2}[/tex] and F_{n+3}-F_{n}=2F_{n+1}[/tex]
Yes, since the first is true for all $n\ge0$, the other two are true for all $n\ge0$.

- Hollywood