$[$Maximum mark: $15]$

Sarah has started a $12$ week training program to improve her cycling. She plans to cycle $100$ kilometres in the first week and, in each subsequent week, cycle $15$ kilometres further than she did the week before.

$($a$)$ Calculate the distance she will cycle during the eighth week.

$($b$)$ Calculate the total distance she will have cycled by the end of the training program.

Sarah's coach tells her she should have cycled $2500$ kilometres by the end of the training program, so Sarah decides to increase her cycling distances. She still plans to cycle $100$ kilometres in the first week, but in each subsequent week, she will cycle $k$ kilometres more than she did the week before.

$($c$)$ Calculate the minimum integer value of $k$ required for Sarah to reach the target of $2500$ kilometres in $12$ weeks.

$3$

As part of her training, each week Sarah performs a one kilometre time trial, in which she aims to cycle one kilometre as fast as she can. Sarah's time decreases by $2\%$ each week. In the first week, her time is exactly $110$ seconds.

$($d$)$ Calculate Sarah's time in the $12$ th week.

$3$

Sarah's goal is to cycle one kilometre in less than $85$ seconds by the end of the training program. To do this, she needs to decrease her time by at least $m\%$ per week from her time of $110$ seconds in week one.

$($e$)$ Find the minimum value of $m$ so that Sarah reaches her goal.

$4$