Principle of Mathematical Induction/Warning/Example 3

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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:

\(\ds 1 + 3 + 5 + \dotsb + \paren {2 k - 1} + \paren {2 k + 1}\) \(=\) \(\ds k^2 + 3 + \paren {2 k + 1}\) 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.


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

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