II$.$ Question $2[$Cash Flow Analysis$]$

Pat is the type to plan everything in advance, and she is already starting to plan for retire$-$

ment. She plans to work for $40$ years, and then retire for the following $30$ years. She expects

to spend $$120,000$ in your first year of retirement, with a $4\%$ growth rate. She will increase

her working$-$years savings rate by $7\mathrm{.}5\%$ per year. While working, she expects her account

will have a $6\mathrm{.}75\%$ return, and during retirement it will have a $4\mathrm{.}75\%$ return. After the $30$ th

year of retirement, she wants to have $$600,000$ in her account for safety net and bequest

reasons $($It is fine if the amount is not exactly $$600,000$ due to rounding error$).$ To solve the

problem, determine how much Pat needs to save in year $1$ to accomplish her retirement goals