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:

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

on the, but:


 * $2^2 + 3 = 7$

on the.

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.