Describe your entire process $($timeline$,$ mathematical formulas and results$),$ not just only the answers. Today is May $18,2024.$ Your grandmother Paulette informs you today that your grandfather Jean$-$Paul is ill and has little chance of recovery. Your grandfather was not a fan of paperwork. His lack of organization means that the follow$-$up on his records is lacking. During the conversation, she mentions that he has several investments in various products and companies but cannot find the actual documents. Since you are taking a finance course at university, she is convinced $($and so are we$!)$ that you will be able to determine how much money your uncle has accumulated to date with his various investments.

Here is the information you find:

On May $18,1965,$ Jean$-$Paul invested $$5,000$ in a renewable certificate of deposit at the bank offering a simple interest rate of $8\%$ per year. The money was intended to pay for his grandchildren$$s education. On May $18,1990,$ after talking to a friend at a New Year's reception, he realized that simple interest was not advantageous and decided to invest this money in another institution offering an annually compounded interest rate of $5\%.$ He never had grandchildren, so this amount was never used and is still in the bank.

On May $18,2000,$ after being absorbed by the Y$2$K bug, Jean$-$Paul started investing $$6,500$ per year for exactly $7$ years in mutual funds. The first year of investment, he faced the full impact of the tech bubble with a return of $-40\%.$ Not losing hope, he continued investing in the markets from $2001$ to $2007$ with an annual return of $15\%$ each year. However, in $2008,$ he was "plucked" with a return of $-65\%.$ Since $2009,$ the returns obtained from his investment have been $17\%$ effective annually until today.

On May $18,1995,$ Jean$-$Paul invested $$250,000$ in a childhood friend$$s private company. In exchange for this amount, his friend promised to pay him an annual dividend $($i$.$e$.,$ income$)$ of $$10,500$ growing by $7\%$ annually from May $18,2001.$ The promise was kept until May $18,2005,$ when Jean$-$Paul received his last dividend due to the company's financial difficulties. On May $18,2007,$ he then received only $$25,000$ of his capital due to the company's closure. Fortunately, Jean$-$Paul always placed his dividends and the $$25,000$ amount in a special account offering an effective annual rate of $12\%$ until today.

A$.$ What is the total amount accumulated in each of the three investments?

Additionally, your grandmother needs to find the best possible place for her husband's eternal rest and wants him to be buried in the cemetery of his hometown. She is told that a plot in this cemetery costs $$10,000$ plus $$500$ per year for "maintenance" $($payable at the beginning of the year$)$ with an increase of $2\mathrm{.}5\%$ per year. The effective annual interest rate to consider is $5\%$ per year. Your aunt finds it a bit expensive for the "maintenance" her husband will actually require but has no choice and buys a plot anyway.

B$.$ What is the value of a plot in this cemetery $($i$.$e$.,$ the present value of the payments to be made$)?$