Assume that you are $20$ years old and started saving for retirement on January $1,2022.$ You plan to retire on December $31,2071,$ when you are $70$ years old. There are $50$ years from the time you started investing $($saving$)$ until you retire. You have no previous or other retirement savings when you start to save. Assume there are $365$ days and $52$ weeks in each year from $2022$ to $2071.($Ignore leap years$).$ Assume that taxes will not affect any of the amounts or your savings.

You invest $$300$ at the end of each month into a retirement account paying $7\%$ compounded monthly from January $1,2022,$ until you retire. Show all work and answer the following questions:

Assuming no withdrawals or additional payments were made, how much money will be in your retirement account after $50$ years?

Assuming you made all the monthly payments for $50$ years, how much did you pay into your retirement account?

Assuming you made all the monthly payments for $50$ years, how much interest did you earn over the $50$ years?