This is a classic retirement problem. Your friend, Mary Jones, is celebrating her $30$th birthday and wants to start saving for her anticipated retirement. She has the following years to retirement and retirement spending goals, which are as follows:

Years until retirement: $30$

Amount to withdraw each year upon retirement: $$90,000$

Years to withdraw in retirement: $20$

Interest rate: $5\%$

Mary is planning to make equal annual deposits into her retirement account, while her first withdrawal will take place one year after her last deposit.

$1.$ If Mary starts making these deposits in one year and makes her last deposit on the day she retires, what amount must she deposit annually to be able to make the desired withdrawals at retirement?

$2.$ What if Mary starts making these deposits when she turns $40$ years old and only has $20$ years left to retirement, what amount must she deposit annually to be able to make the same desired withdrawals?

$3.$ Suppose Mary $($on her $30$th birthday$)$ has just inherited a large sum of money. Rather than making equal annual payments, she has decided to make one lump sum deposit today to cover the entire cost of her retirement, how much would she need to deposit today to cover the anticipated withdrawals?

$4.$ Assume the same scenario as No$.3$ above, but the interest rate she can earn over the next $30$ years is only $3\%$; what would she need to deposit today to cover the anticipated withdrawals?

$5.$ When you compare the results of No$.1$ and No$.2$ above, what is the key factor that causes the results to be so different?