A cycle is an even permutation iff its length is odd

Question

Prove that for all pairwise distinct $a_1,a_2,\ldots,a_t \in \{1,2,\ldots,n\}, (a_1,a_2, \ldots a_t) \in A_n$ if and only if $t$ is odd.

How to prove the odd cycles of transposition will be product of even number of transpositions in this case

Answer 1

$(123\dots n)=(1n)(1n-1)\dots(12)$ for instance. There's nothing special about $1,2,\dots,n$ here.Thus a cycle of odd length is even.