91Ó°ÊÓ

Write down the generating function for the number of ways to partition an integer into parts of size no more than \(m,\) each used an odd number of times. Write down the generating function for the number of partitions of an integer into parts of size no more than \(m,\) each used an even number of times. Use these two generating functions to get a relationship between the two sequences for which you wrote down the generating functions. (h)

Suppose we deposit \(\$ 5000\) in a savings certificate that pays ten percent interest and also participate in a program to add \(\$ 1000\) to the certificate at the end of each year (from the end of the first year on) that follows (also subject to interest.) Assuming we make the \(\$ 5000\) deposit at the end of year 0 , and letting \(a_{i}\) be the amount of money in the account at the end of year \(i,\) write a recurrence for the amount of money the certificate is worth at the end of year \(n\). Solve this recurrence. How much money do we have in the account (after our year-end deposit) at the end of ten years? At the end of 20 years?