# Upper Bound for Lucas Number

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## Contents

## Theorem

Let $L_n$ denote the $n$th Lucas number.

Then:

- $L_n < \paren {\dfrac 7 4}^n$

## Proof

The proof proceeds by complete induction.

For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:

- $L_n < \paren {\dfrac 7 4}^n$

$\map P 1$ is the case:

\(\displaystyle L_1\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \dfrac 7 4\) |

Thus $\map P 1$ is seen to hold.

### Basis for the Induction

$\map P 2$ is the case:

\(\displaystyle L_2\) | \(=\) | \(\displaystyle 3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {48} {16}\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \dfrac {49} {16}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\dfrac 7 4}^2\) |

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that if $\map P j$ is true, for all $j$ such that $0 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.

This is the induction hypothesis:

- $L_k < \paren {\dfrac 7 4}^k$

from which it is to be shown that:

- $L_{k + 1} < \paren {\dfrac 7 4}^{k + 1}$

### Induction Step

This is the induction step:

\(\displaystyle L_{k + 1}\) | \(=\) | \(\displaystyle L_k + L_{k - 1}\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \paren {\dfrac 7 4}^k + \paren {\dfrac 7 4}^{k - 1}\) | Induction Hypothesis | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\dfrac 7 4}^{k - 1} \paren {1 + \dfrac 7 4}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\dfrac 7 4}^{k - 1} \paren {\dfrac {11} 4}\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \paren {\dfrac 7 4}^{k - 1} \paren {\dfrac 7 4}^2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\dfrac 7 4}^{k + 1}\) |

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

Therefore:

- $\forall n \in \Z_{\ge 1}: L_n < \paren {\dfrac 7 4}^n$

$\blacksquare$

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Example $1 \text{-} 1$