A closed-form expression will be a function that will give you a value from one calculation using an input, while a recursion requires you to begin with some initial value and then to compute all successive values up to the input.

1.) We find:

2.) We may now state:

(1)

This is a linear inhomogeneous recursion. To find the closed-form, we may state:

(2)

Now we may use symbolic differencing to obtain a homogeneous recursion. Subtracting (1) from (2) we find:

Now we have a homogeneous recursion, whose associated characteristic equation is:

Given the characteristic roots, the closed-form is then:

Using the initial value we computed in part 1.), we may determine that parameters :

Solving this system, we find:

hence:

Now, you may use this to find , the amount Bill has at his 45th birthday.

3.) See if you can apply the above technique, using the result from part 2.) as your starting value, and you will begin with a homogeneous recursion.