$5\mathrm{.}115$ A couple wants to accumulate $$10000$ by making deposits of $$800$ at the end of each half$-$year into a savings account that earns interest at $j_2=9\%.($a$)$ How many full deposits must they make? $($b$)$ What additional deposit made at the time of the last full deposit will be necessary to bring the savings account to $$10000?($c$)$ What deposit made $6$ months after the last full deposit will bring the account to $$10000?$

Ans. $($a$)10$; $($b$)$ $$169\mathrm{.}43$; $($c$)$ no deposit necessary $($account has $$10272\mathrm{.}94)$