Lucas Number as Element of Recursive Sequence
Theorem
Let $L_k$ be the $k$th Lucas number, defined as the sum of two Fibonacci numbers:
- $L_n = F_{n - 1} + F_{n + 1}$
Then $L_n$ can be defined as the $n$th element of the recursive sequence:
- $L_n = \begin{cases}
2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
Proof
Proof by induction:
Let $L_n$ be the Lucas number defined as the sum of two Fibonacci numbers:
- $L_n = F_{n - 1} + F_{n + 1}$
For all $n \in \N$, let $\map P n$ be the proposition:
- $L_n = \begin{cases}
2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
Basis for the Induction
We have that:
- $L_0 = F_{-1} + F_1 = 1 + 1 = 2$
- $L_1 = F_0 + F_2 = 0 + 1 = 1$
- $L_2 = F_1 + F_3 = 1 + 2 = 3$
Thus $\map P 0$, $\map P 1$ and $\map P 2$ hold.
$\map P 3$ is the case:
- $L_3 = F_2 + F_4 = 1 + 3 = 4$
So $\map P 3$, as $L_3 = L_1 + L_2$.
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:
- $\forall 1 \le j \le k: L_j = L_{j - 1} + L_{j - 2}$
Then we need to show:
- $L_{k + 1} = L_k + L_{k - 1}$
Induction Step
This is our induction step:
\(\ds L_{k + 1}\) | \(=\) | \(\ds F_k + F_{k + 2}\) | by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{k - 2} + F_{k - 1} + F_k + F_{k + 1}\) | Definition of Fibonacci Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {F_{k - 2} + F_k} + \paren {F_{k - 1} + F_{k + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds L_{k - 1} + L_k\) | by hypothesis |
Hence $L_n = L_{n - 2} + L_{n - 1}$ follows by the Second Principle of Mathematical Induction.
That is: $\sequence {L_n}$ is the sequence defined as:
- $L_n = \begin{cases}
2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Exercise $13$