Lucas Number as Sum of Fibonacci Numbers
Theorem
Let $L_k$ be the $k$th Lucas number, defined as:
- $L_n = \begin{cases}
2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
Then:
- $L_n = F_{n - 1} + F_{n + 1}$
where $F_k$ is the $k$th Fibonacci number.
Proof
Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
- $L_n = F_{n - 1} + F_{n + 1}$
Basis for the Induction
$\map P 1$ is true, as this just says:
- $L_1 = 1 = F_0 + F_2$
which holds by definition of the Fibonacci numbers.
This is our basis for the induction.
Induction Hypothesis
Let us make the supposition that, for some $k \in \N: k \ge 1$, the proposition $\map P j$ holds for all $j \in \N: 1 \le j \le k$.
We shall show that it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
- $L_j = F_{j - 1} + F_{j + 1}$
Then we need to show:
- $L_{k + 1} = F_k + F_{k + 2}$
Induction Step
This is our induction step:
\(\ds L_{k + 1}\) | \(=\) | \(\ds L_{k - 1} + L_k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds F_{k - 2} + F_k + F_{k - 1} + F_{k + 1}\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {F_{k - 2} + F_{k - 1} } + \paren {F_k + F_{k + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds F_k + F_{k + 2}\) |
So $\map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.
Therefore:
- $\forall n \in \N, n \ge 1: L_n = F_{n - 1} + F_{n + 1}$
$\blacksquare$