Honsberger's Identity/Proof 1

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 $\map P n$ be the proposition:

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

Basis for the Induction

$\map P 1$ is the case:

\(\ds F_{m + 1}\)

\(=\)

\(\ds F_{m - 1} + F_m\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds F_{m - 1} \times 1 + F_m \times 1\)

\(\ds \)

\(=\)

\(\ds F_{m - 1} F_1 + F_m F_2\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds F_{m - 1} F_n + F_m F_{n + 1}\)

for $n = 1$

and so $\map P 1$ is seen to hold.

$\map P 2$ is the case:

\(\ds F_{m + 2}\)

\(=\)

\(\ds F_{m + 1} + F_m\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds F_{m - 1} + F_m + F_m\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds F_{m - 1} \times 1 + F_m \times 2\)

\(\ds \)

\(=\)

\(\ds F_{m - 1} F_2 + F_m F_3\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds F_{m - 1} F_n + F_m F_{n + 1}\)

for $n = 2$

and so $\map P 2$ is seen to hold.

This is our basis for the induction.

Induction Hypothesis

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

So this is our induction hypothesis:

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

and:

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

from which it is to be shown:

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

Induction Step

This is our induction step:

\(\ds F_{m + k + 1}\)

\(=\)

\(\ds F_{m + k} + F_{m + k - 1}\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds F_{m - 1} F_k + F_m F_{k + 1} + F_{m - 1} F_{k - 1} + F_m F_k\)

Induction Hypothesis

\(\ds \)

\(=\)

\(\ds F_{m - 1} \paren {F_k + F_{k - 1} } + F_m \paren {F_{k + 1} + F_k}\)

\(\ds \)

\(=\)

\(\ds F_{m - 1} F_{k + 1} + F_m F_{k + 2}\)

Definition of Fibonacci Number

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

Therefore:

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

$\blacksquare$