Charlotte is negotiating a prenuptial agreement with Bunny. Under the agreement,

Charlotte will be receiving $10\%$ of $$500$K every three years $($with the first payment

occurring three years after the wedding$).$ Bunny claims that under the agreement, and

provided that the marriage lasts for $30$ years, Charlotte is worth $$500$K$.$

$($a$)$ Charlotte has some doubts, and talks it over with Miranda, who has studied

some finance during her law school education. Miranda claims that under the

agreement, Charlotte is worth less than $$500$K$.$ Why is Miranda right, and what

is her argument?

$($b$)$ Miranda advises Charlotte to negotiate an agreement under which she is worth

$$500$K$.$ What should each of the $10($equal$)$ payments be$,$ provided that Charlotte$$s

annual discount rate is $4\%?$