Planning Investments. Your uncle has $$90,000$ that he wishes to invest now in order to use the accumulation for purchasing a retirement annuity in five years. After consulting with his financial adviser, he has been offered four types of fixed income investments, labeled as investments A$,$ B$,$ C$,$ and D$.$

Investments A and B are available at the beginning of each of the next five years $($call them years $1$ to $5).$ Each dollar invested in A at the beginning of a year returns $$1\mathrm{.}20($a profit of $$0\mathrm{.}20)$ two years later, in time for immediate reinvestment. Each dollar invested in B at the beginning of a year returns $$1\mathrm{.}36$ three years later.

Investments C and D will each be available at one time in the future. Each dollar invested in C at the beginning of year $2$ returns $$1\mathrm{.}66$ at the end of year $5.$ Each dollar invested in D at the beginning of year $5$ returns $$1\mathrm{.}12$ at the end of year $5.$

Your uncle is obligated to make a balloon payment on an existing loan, in the amount of $$24,000,$ at the end of year $3.$ He wants to cover that payment out of these funds as well.

a$.$ Devise an investment plan for your uncle that maximizes the amount of money that can be accumulated at the end of five years. How much money will be available for the annuity in five years?

b$.$ Describe the pattern in the optimal plan.