2017 NYJC Promo Q12

Two athletes signed up for the \(13\) km Nanyang Marathon. In preparation for this marathon, the athletes planned a \(16\)-day personalised training programme as follows:

Athlete \(A\) will run \(0.4\) km on the first day of the training programme and for each subsequent day, the distance he will run is increased by \(x\) km.

Athlete \(B\) will run \(0.6\) km on the first day of the training programme and for each subsequent day, the distance he will run is \(10\%\) more than the distance ran for the previous day.

1. Find the minimum value of \(x\) such that Athlete \(A\) is able to run at least \(13\) km on the last day of the training programme.[2]
2. Athlete \(B\) targets to run a total distance of \(26\)km in these \(16\) days. Can he achieve this target? You must show sufficient workings to justify your answer.[2]

After \(2\) days of training, Athlete \(B\) modified his training programme such that on Day \(n\), \(3\le n\le 16\), he will run \(\left[ \frac{10}{21}{{\left( 1.05 \right)}^{n}}+n \right]\)km. There are no further changes to his training programme.

3. Show that the total distance Athlete \(B\) will run by the end of Day \(m\), where \(3\le m\le 16\), can be expressed as
   \(P\left( {{Q}^{m}} \right)+\frac{\left( m-2 \right)\left( m+3 \right)}{2}+R\),
   where \(P\), \(Q\) and \(R\) are exact constants to be determined.[5]
4. Athlete \(B\) feels that running more than \(13\) km per training session may be too strenuous. Assuming he follows the new training programme but limit his running distance to \(13\) km per training session, find the total distance he will run in the \(16\) days of training.[4]