# Principle of Mathematical Induction/Warning/Example 3

Jump to navigation
Jump to search

## Example of Incorrect Use of Principle of Mathematical Induction

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.

## Refutation

The basis for the induction has not been established.

It is in fact not possible to find a value of $n$ for which $1 + 3 + 5 + \dotsb + \paren {2 n - 1} = n^2 + 3$ actually holds.

$\blacksquare$

## Sources

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