David wants to open a small Nordic$-$style spa in $4$ years$$ time. The building he wants to buy for this project is worth $$18,500,000.$ He knows that to open the spa he$$ll need a $20\%$ down payment for the building plus an additional $$2,500,000$ for renovations and decor.

$1.$ David can secure a high$-$yield investment that earns $12\mathrm{.}3\%$ compounded semi$-$annually, and he plans to make a deposit in the account at the beginning of every three months. How much will he have to deposit every quarter if he wants to open the spa in $4$ years?

To take possession of the building, David pays the down payment and secures a mortgage for the remaining balance at an interest rate of $4\mathrm{.}9\%$ compounded monthly for $25$ years.

$2.$ If the interest rate is constant over the $25-$year term, and payments are made at the end of every two weeks, how much are David$$s biweekly payments? What will be the total interest paid on the spa over the term?

When the renovations to the spa are finished $($including the name $$Tickle Me Pink$$ in stylish lettering on the front$),$ David is pleased to find that they were completed under$-$budget, costing him only $$1,940,000$ instead of the $$2,500,000$ he originally planned.

$3.$ David invests the surplus in another high$-$yield investment that earns $9\mathrm{.}8\%$ compounded daily. How much is this investment worth in $8$ years$$ time?

After $8$ years of owning the spa, David$$s investment $($from Question $3)$ matures and he decides to use this balance to build an addition to his spa. He makes weekly withdrawals from his investment account to finance this new venture.

$4.$ David withdraws $$6,000$ at the end of every week for the first $3$ years to establish and market the expansion. Afterwards, the withdrawals will lower to $$3,500$ at the end of every week. Throughout this period, the high$-$yield investment continues to earn $9\mathrm{.}8\%$ compounded daily. What is the total length of time that David will be able to make weekly withdrawals for? Express your answer in years and months.

Decades later, several years after the mortgage on the spa is paid off, David decides to sell the spa and retire. He wants to purchase a mansion on the coast of the Mediterranean, and also invest in a fund that will allow him to live comfortably on $$45,000$ at the beginning of each quarter for the rest of his life.

$5.$ If the mansion costs $$3,200,000$ to purchase outright $($with no mortgage$),$ and David can secure an investment that earns $2\mathrm{.}1\%$ compounded monthly, what is the minimum amount he can accept for the sale of the spa?

$6.$ If David sells his spa for $$10,000,000,$ what will be the size of his perpetual quarterly payments instead?