Fibonacci Number in terms of Smaller Fibonacci Numbers/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$


Proof

From the initial definition of Fibonacci numbers, we have:

$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

Proof by induction:

For all $n \in \Z_{>0}$, let $P \left({n}\right)$ be the proposition:

$\displaystyle \forall m \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$


Basis for the Induction

$P \left({1}\right)$ is the case:

\(\displaystyle F_{m + 1}\) \(=\) \(\displaystyle F_{m - 1} + F_m\) Definition of Fibonacci Number
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} \times 1 + F_m \times 1\)
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} F_1 + F_m F_2\) Definition of Fibonacci Number
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} F_n + F_m F_{n + 1}\) for $n = 1$

and so $P \left({1}\right)$ is seen to hold.


$P \left({2}\right)$ is the case:

\(\displaystyle F_{m + 2}\) \(=\) \(\displaystyle F_{m + 1} + F_m\) Definition of Fibonacci Number
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} + F_m + F_m\) Definition of Fibonacci Number
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} \times 1 + F_m \times 2\)
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} F_2 + F_m F_3\) Definition of Fibonacci Number
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} F_n + F_m F_{n + 1}\) for $n = 2$

and so $P \left({2}\right)$ is seen to hold.


This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ and $P \left({k-1}\right)$ are true, where $k > 1$, then it logically follows that $P \left({k + 1}\right)$ is true.


So this is our induction hypothesis:

$\displaystyle F_{m + k} = F_{m - 1} F_k + F_m F_{k + 1}$

and:

$\displaystyle F_{m + k - 1} = F_{m - 1} F_{k - 1} + F_m F_k$


from which it is to be shown:

$\displaystyle F_{m + k + 1} = F_{m - 1} F_{k + 1} + F_m F_{k + 2}$


Induction Step

This is our induction step:

\(\displaystyle F_{m + k + 1}\) \(=\) \(\displaystyle F_{m + k} + F_{m + k - 1}\) Definition of Fibonacci Number
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} F_k + F_m F_{k + 1} + F_{m - 1} F_{k - 1} + F_m F_k\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} \left({F_k + F_{k - 1} }\right) + F_m \left({F_{k + 1} + F_k}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle F_{m - 1} F_{k + 1} + F_m F_{k + 2}\) Definition of Fibonacci Number

So $P \left({k}\right) \land P \left({k - 1}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\displaystyle \forall m, n \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

$\blacksquare$