You have just turned $30$ years old, have just received your MBA, and have accepted your first job. Now you must decide how much money to put into your retirement plan. The plan works as follows: Every dollar in the plan earns $7\%$ per year. You cannot make withdrawals until you retire on your sixty$-$fifth birthday. After that point, you can make withdrawals as you see fit. You decide that you will plan to live to $100$ and work until you turn $65.$ You estimate that to live comfortably in retirement, you will need $$100,000$ per year starting at the end of the first year of retirement and ending on your $100$th birthday. You will contribute the same amount to the plan at the end of every year that you work. How much do you need to contribute each year to fund your retirement?

$1.$ Solve it analytically, that is$,$ use formulas to derive and calculate annual contributions.

$2.$ Use Excel commands to solve it$.$ How does bank balance evolve through time $($year$-$by$-$year from age $35$ to $100)?$ Build a graph of bank balance $($balance on Y$-$axis, year on X$-$axis$).$