# Principle of Mathematical Induction/Warning/Example 3

## 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.

$\blacksquare$