Annuities

$($a$)$ Find the final amount in the following retirement account, in which the rate of return on the account and the regular contribution change over time. $$623$ per month invested at $5\%,$ compounded monthly, for $7$ years; then $$787$ per month invested at $7\%,$ compounded monthly, for $7$ years. What is the amount in the account after $14$ years?

$($b$)$ Find the amount of each payment to be made into a sinking fund earning $7\%$ compounded monthly to accumulate $$15,000$ over $8$ years. Payments are made at the end of each period.

$($c$)$ Suppose a set of new parents wants to have some money saved up to help their child with college. They'd like to have enough money to fund equal yearly payments of $$15,000$ for the four years of college. They have an account that earns $4\mathrm{.}4\%$ compounding annually. How much should they have in the account when their child turns $18?$

$($d$)$ Suppose the same new parents start putting money into the same account each year in order to get the amount in part $($h$).$ How much should they put in each year from the time the child is born until the time the child turns $18?$

$($e$)$ Hank wants to retire when he has enough money in his account, which earns $3\mathrm{.}4\%$ interest compounded monthly, to fund monthly payments of $$2000$ for $10$ years. How much should he have when he retires?