$1.$ You are given that a$($t$)$ is of the form bt$2+$ c$.$ If $$100$ invested at time $0$ accumulates to $$172$

at time $3.$ Note this is a special way of accumulating interest that we did not discuss in class.

$($a$)$ Interpret a$($t$).$ Determine the value of b and c$,$ respectively.

$($b$)$ Find the present value of $$500$ paid at time $5.$