I have observed for first few values of consecutive odd numbers, the result is always of the form:
$m^2 - n^2$, where $m$ and $n$ are two distinct positive integers. That is: $$1\cdot 3\cdot 5 \cdot 7\cdots (2k-1) = m^2 - n^2$$
For example:
$ 1\cdot 3 = 3 = 2^2 - 1^2 \\ 1\cdot 3\cdot 5 = 15 = 4^2 - 1^2 \\ 1\cdot 3\cdot 5\cdot 7 = 105 = 11^2 - 4^2 \\ 1\cdot 3\cdot 5\cdot 7\cdot 9 = 945 = 31^2 - 4^2 \\ \vdots $
But not sure, how to prove it. Here is an attempt using induction:
Let it be true for some value of k, that is: $1\cdot 3\cdot 5\cdot 7\cdot 9\cdots(2k - 1) = m^2 - n^2$
Then when $k$ takes the value of $k+1$, we have
$$\begin{align} 1\cdot 3\cdot 5\cdot 7\cdot 9\cdots(2k - 1)\cdot(2k + 1) &= (m^2 - n^2)\cdot(2k + 1)\\ &= (m^2 - n^2)\cdot {(k+1)^2 - k^2} \end{align}$$ and got stuck here.
Can you please suggest to proceed further or an altogether a different way of proving so or prove me wrong.
Thanks in advance.