Principle of Mathematical Induction/Warning
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Principle of Mathematical Induction: Warning
It is a common mistake when setting up a proof by induction to forget to check the base case.
This can cause incorrect reasoning.
Example 1
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:
\(\ds L_{k + 1}\) | \(=\) | \(\ds L_k + L_{k - 1}\) | Definition 1 of Lucas Number | |||||||||||
\(\ds \) | \(=\) | \(\ds F_k + F_{k - 1}\) | by assumption | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{k + 1}\) | Definition of Fibonacci Number |
Hence:
- $\forall n \in \Z_{>0}: F_n = L_n$
Example 2
We are to prove that:
- $\dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 3} + \dotsb + \dfrac 1 {\paren {n - 1} \times n} = \dfrac 3 2 - \dfrac 1 n$
For $n = 1$ we have:
- $\dfrac 3 2 - \dfrac 1 n = \dfrac 1 2 = \dfrac 1 {1 \times 2}$
Assuming true for $k$, we have:
\(\ds \dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 3} + \dotsb + \dfrac 1 {\paren {n - 1} \times n} + \dfrac 1 {n \times \paren {n + 1} }\) | \(=\) | \(\ds \dfrac 3 2 - \frac 1 n + \dfrac 1 {n \paren {n + 1} }\) | by the induction hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 3 2 - \frac 1 n + \paren {\dfrac 1 n - \dfrac 1 {n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 3 2 - \frac 1 {n + 1}\) |
But clearly this is wrong, because for $n = 6$:
- $\dfrac 1 2 + \dfrac 1 6 + \dfrac 1 {12} + \dfrac 1 {30} = \dfrac 5 6$
on the left hand side, but:
- $\dfrac 3 2 - \dfrac 1 6 = \dfrac 4 3$
on the right hand side.
Example 3
We are to prove that:
- $1 + 3 + 5 + \dotsb + \paren {2 n - 1} = n^2 + 3$
We establish as an induction hypothesis:
- $1 + 3 + 5 + \dotsb + \paren {2 k - 1} = k^2 + 3$
Then:
\(\ds 1 + 3 + 5 + \dotsb + \paren {2 k - 1} + \paren {2 k + 1}\) | \(=\) | \(\ds k^2 + 3 + \paren {2 k + 1}\) | from the induction hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds k^2 + 2 k + 1 + 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k + 1}^2 + 3\) | Square of Sum |
But clearly this is wrong, because for $n = 2$, say:
\(\ds \paren {2 \times 1 - 1} + \paren {2 \times 2 - 1}\) | \(=\) | \(\ds 1 + 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4\) |
on the left hand side, but:
- $2^2 + 3 = 7$
on the right hand side.
Also see
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction