and verify it has the solution
There are two ways you can do this:
1) Start with the solution and verify it fits the equation. (Work backwards, in other words.)
2) Start with the equation for and work your way down, step by step.
The first method is practically trivial and probably not what you want so I'll work the second one.
Putting this into your first expression gives:
Again, . Put this into the equation. You'll get
Notice the pattern. At the "a"th step we get
So at the nth step, we get:
your desired solution.