Why is this simple list of products not simplified?

Why is this product ratio not simplified to a[n]?

Clear[a,k,n]
$ Assumptions = {n> 1}
Simplify[ Product[a[k]{k, 1, n}]/ Product[a[k]{k, 1, n - 1}]]

Establishing a real value of $ n $, works well, for example, set n = 5 and you get a[5].
If you define a function to[k] It seems that sometimes it works well. It does not work for

Product[1-1/Sqrt[1-1/Sqrt[1-1/Sqrt[1-1/Sqrt[k]{k, 2, n}]/
Product[1-1/Sqrt[1-1/Sqrt[1-1/Sqrt[1-1/Sqrt[k]{k, 2, n – 1}]