Assuming you dollars at the start of that year, let's try to find the amount we have at the end of the year:

at the end of each quarter, we multiply by 1.057,

hence, at the end of the year we have: . Our goal is to find n for which .

This is difference equation which looks like: whose solution is of the form (A, r and C are constants). To find these, solve first for . You get A and r. Then plug in what you got in the initial equation to find C.

Now that you have the closed form, you can solve for

hope this helps