You have $$10,000$ you want to invest for the next $20$ years. You are offered an

investment plan that will pay $8\%$ per year for the next $10$ years and $12\%$ per year for

the last $10$ years. How much will you have at the end of the $20$ years? Does it matter if

the investment plan pays you $12\%$ per year for the first $10$ years and $8\%$ per year for

the next $10$ years? Why or why not? If another investment plan pays $10\%$ per year for

the next $20$ years, which investment plan is better? Consider two investment outcome

schemes: $(1)$ Interest income will be reinvested at each year. $(2)$ Interest income will

be paid out at each year. Provide an intuitive answer and also a numerical answer.