Principle of Mathematical Induction/Warning/Example 1

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Example of Incorrect Use of Principle of Mathematical Induction

Let $L_k$ denote the $k$th Lucas number.

Let $F_k$ denote the $k$th Fibonacci number.


Given that $L_n = F_n$ for $n = 1, 2, \ldots, k$, we see that:

\(\displaystyle L_{k + 1}\) \(=\) \(\displaystyle L_k + L_{k - 1}\) Definition 1 of Lucas Number
\(\displaystyle \) \(=\) \(\displaystyle F_k + F_{k - 1}\) by assumption
\(\displaystyle \) \(=\) \(\displaystyle F_{k + 1}\) Definition of Fibonacci Number


Hence:

$\forall n \in \Z_{>0}: F_n = L_n$


Refutation

We have made the assumption that $L_n = F_n$ for $n = 1, 2, \ldots, k$.

However, we have that:

$L_2 = 3$

while:

$F_2 = 1$

Hence as the base case has been demonstrated to be false, the proof is invalid.

$\blacksquare$


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