$8$

Julie is planning her retirement. She is $30$ and will invest for the next $30$ years in her retirement account. She expects the retirement account to yield $8\%$ a year in returns. Assuming she invests $$10,000$ a year in the beginning of each year $($and interest is therefore obtained in the investment on that year$),$ how much will she have at the end of the $30$ year period?

$(10$ Points$)$

The value must be a number

$$

She is not happy with the amount as she thinks she will need $2$ million dollars to retire. How much money would she need to invest each year $($instead of $$10,000)$ to obtain $2$ million dollars at the end of $30$ years? $(10$ Points$)$

The value must be a number

$10$

That amount was too much for Julie. She has an alternative plan. She now wants to start with $$10,000$ she can afford and then increase the investment amount by $2\%$ a year $($she expects her salary will increase and she will be able to deposit more$).$ For example, on year $1,$ she deposits $$10,000$; on year $2,$ she deposits $$10,{000}^{**}(1\mathrm{.}02)=$$$10,200$; on year $3,$ she deposits