Billy contracted a loan at the bank. He pays back his loan in $3$ payments : the first one

is $$5000$ made after one year, the second one of $$7000$ made one year and a half after

he contracted the loan, and the last one, of $$3000,$ made three years after he contracted

the loan. The interest rate is $i^{(2)}=10\%.$

$($a$)$ Determine the amount of the loan.

$($b$)$ If after his first payment of $$5000,$ Billy renegotiates his loan with the bank such

that he only has one payment left of $$x$ to do$,$ four years after he contracted the

loan, with the same interest rate, what should be the value of $x?$

$($c$)$ Consider the situation in b$).$ If Billy decides to accumulate $$x$ in a separate bank

account, by doing $36$ monthly payments of $$P,$ with a nominal interest rate of

$i^{(12)}=6\%,$ compounded each month. The first payment will be one month after

his payment of $$5000.$ Determine $P,$ using annuities.