Is it even odd?

Number Theory Level 2

Let a series be defined as $a_0=1$ $a_n=\frac{n!}{a_{n-1}},n>0$ .

Now consider the following statements.

All are true Only 3 is false All are false Only 2 is false 2 and 3 are not true.

1 solution

Janardhanan Sivaramakrishnan
Jul 8, 2016

Consider the relation between $a_{2n+2}$ and $a_{2n}$ .

$a_{2n+2}=\frac{(2n+2)!}{\frac{(2n+1)!}{a_{2n}}}=2(n+1)a_{2n}=2^2(n+1)na_{2n-2}$

Using the fact that $a_0=1$ , we can conclude that $a_{2n}=2^nn!$ , which is always an even integer for $n>0$ . Thus confirming statement 3.

Similarly, $a_{2n+1}=(2n+1) a_{2n-1}$ . Noting that $2n+1$ is always odd and $a_1=\frac{1!}{a_0}=1$ , statement 2 is also confirmed.

Statement 1 is a trivial conclusion, if statements 2 and 3 are true.