Lucas Number as Sum of Fibonacci Numbers

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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:

\(\displaystyle L_{k + 1}\) \(=\) \(\displaystyle L_{k - 1} + L_k\)
\(\displaystyle \) \(=\) \(\displaystyle F_{k - 2} + F_k + F_{k - 1} + F_{k + 1}\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \paren {F_{k - 2} + F_{k - 1} } + \paren {F_k + F_{k + 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle F_k + F_{k + 2}\)

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


Therefore:

$\forall n \in \N, n \ge 1: L_n = F_{n - 1} + F_{n + 1}$

$\blacksquare$