$($a$)$ Show from first principles that $|)=\frac{s_{\frac{?}{\text{b}\text{a}\text{r}}()}^{}-n}{},$ where $$ is constant for

the $n$ years.

$($b$)$ The force of interest $(t)$ is a linear function of time and has a value of

$0\mathrm{.}12$ at time $0$ decreasing to $0\mathrm{.}02$ at time $20,$ whereafter it remains

constant.

A series of continuous payments are purchased, where the payments for the first

$20$ years are at the rate of $(72-3t)$ per annum at time $t$; and those for the last

$15$ years are at rate $r$ per annum, where $r$ denotes the year of payment

measured since time $0.$

Calculate the accumulated amount at time $35$ of these continuous payments.