# Principle of Mathematical Induction/Warning

## 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:

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

Hence:

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

### 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:

 $\displaystyle \dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 3} + \dotsb + \dfrac 1 {\paren {n - 1} \times n} + \dfrac 1 {n \times \paren {n + 1} }$ $=$ $\displaystyle \dfrac 3 2 - \frac 1 n + \dfrac 1 {n \paren {n + 1} }$ by the induction hypothesis $\displaystyle$ $=$ $\displaystyle \dfrac 3 2 - \frac 1 n + \paren {\dfrac 1 n - \dfrac 1 {n + 1} }$ $\displaystyle$ $=$ $\displaystyle \dfrac 3 2 - \frac 1 {n + 1}$

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:

 $\displaystyle 1 + 3 + 5 + \dotsb + \paren {2 k - 1} + \paren {2 k + 1}$ $=$ $\displaystyle k^2 + 3 + \paren {2 k + 1}$ from the induction hypothesis $\displaystyle$ $=$ $\displaystyle k^2 + 2 k + 1 + 3$ $\displaystyle$ $=$ $\displaystyle \paren {k + 1}^2 + 3$ Square of Sum

But clearly this is wrong, because for $n = 2$, say:

 $\displaystyle \paren {2 \times 1 - 1} + \paren {2 \times 2 - 1}$ $=$ $\displaystyle 1 + 3$ $\displaystyle$ $=$ $\displaystyle 4$

on the left hand side, but:

$2^2 + 3 = 7$

on the right hand side.