I know that nobody cares, but here’s my solution anyways.
This problem defines a linear recurrence such that:
Replacing with a constant in order to find the characteristic polynomial, :
Thus the characteristic polynomial of this linear recurrence is:
Rearrange, and divide by .
This is our characteristic polynomial. Factoring it:
Difference of cubes.
Combine like terms and factor .
The roots of the characteristic polynomial are . The root has a multiplicity of two, so the general solution to the recurrence is given by:
We can plug in the initial values given in the problem, , , , and to find the constants.
Solving this system of three linear equations gives the following values for , , and :
Therefore, the formula for the th term of this sequence is given by:
And, finally, is:
Hey, Andy, you must have solved it intuitively (instead of actually solving the recurrence and finding the general formula), could ya tell me/us how? =D