The first thing that came to mind:

    list1 = {a1, a2, a3, a4, a5};
    
    Rest@FoldList[Append, {}, Most@list1]
    
    Rest[list1] / %

>     {{a2/a1}, {a3/a1, a3/a2}, {a4/a1, a4/a2, a4/a3}, {a5/a1, a5/a2, a5/a3, a5/a4}}

I am still thinking about a *clean* way to perform the second step.