a$.$ Your objective is to have $$1,000,000$ in your account at the end of $25$ years. For the first $5$ years, you expect the interest rate to be $6\%,$ then $8\%$ for the next $10$ years, and $7\%$ for the remaining years. How much do you need to put into the account initially?

b$.$ What$$s the value of a perpetuity that pays $$100$ every $3$ months forever? The discount rate quoted on an APR basis is $6\%.$