Adding Fibonacci Numbers

Question

I am getting confused on adding Fibonacci numbers. For example I know that:

$\mathrm{F}_\mathrm{K+1}+\mathrm{F}_\mathrm{K}=\mathrm{F}_\mathrm{K+2}$

But I believe my logic is flawed. The way I am thinking about this is that if you add $\mathrm{F}_\mathrm{K+1}$ to $\mathrm{F}_\mathrm{K}$ it would become $\mathrm{F}_\mathrm{K+2}$ because you have added a $\mathrm{F}_\mathrm{K}$. I do not understand why or how the following is true
$\mathrm{F}_\mathrm{K}+\mathrm{F}_\mathrm{K-1}=\mathrm{F}_\mathrm{K+1}$

Answer 1

The efinition of Fibonacci numbers is: $$ \begin{align} F_{k} &= F_{k-1} + F_{k-2}, k \ge 2 \\ F_0 &= 0 \\ F_1 &= 1 \end{align} $$ where $k$ is an integer.

Everything else you write about them can be inferred from that.

Of course, assuming $K \ge 0$, writing $F_{K+2} = F_{K+1} + F_K$ is equivalent to above, which means it's right simply by definition. Your second statement, however, is wrong.

Answer 2

Let $l=k-1$. Then $F_{k+2} = F_{k+1} + F_k$ is the same as $F_{l+1}=F_l + F_{l-1}$.

When you are confused by something like this, it is usually best to try specific values. For example, why is $F_{3-1} = F_{3} + F_{3-1}$ true?

Answer 3

Let me restate what you know. You know that $$ F_{k+1} + F_k = F_{k+2} \tag{1} $$ for any integer $k$. That means that in particular, if $l$ is any integer, you can let $k = l - 1$ and you get $$ F_{(l-1) + 1} + F_{(l-1)} = F_{(l-1) + 2} $$ which simplifies to $$ F_l + F_{l-1} = F_{l+1} $$ Since the variable name doesn't matter, we can replace $l$ with $k$, as long as we keep in mind that this $k$ is different than the $k$ we had before. So we get $$ F_k + F_{k-1} = F_{k+1}. \tag{2} $$

In summary: The identity you don't understand (2) is just the same thing as (1), except it is shifted over by one.

Answer 4

Lets make it clear what "$F_n$" denotes. n represents which term it is in the sequence. For example, $F_6$ is the sixth term in the sequence. $F_{n+1}$ means the term $1$ after the $n $th term in the sequence.

Also, it is vital that we understand what the Fibonacci series is:

Each term is equal to the sum of the previous two terms

$F_{n+2}=F_{n+1}+F_n $ just means that the sum of a term and the term directly after is the term after that one.

You stated that $F_n + F_{n-1}=F_{n+1}$

That is correct.

In word form, it says this term add the previous one is the after it.