Caleb is training for a vertical marathon by running up the stairs of a building with 62 floors. He decides to train three days a week, starting with 10 floors on his first day of training. On each subsequent training day, he runs 2 floors more than on the previous training day.
Find the number of floors he completes on his 10th training day.[2]
Once he can complete 62 floors in a training day, he is considered ready for the vertical marathon. On subsequent training days, he will continue to complete 62 floors each day.
He is ready for the vertical marathon at the end of \(k\) weeks of his training. Find \(k\).[2]
Find the total number of floors he completes at the end of 20 weeks of his training.[3]
After training, Caleb’s heart rate decreases by 10% at the start of every \(n\)th minute. When \(n=1\), his heart rate at the start of the first minute is 180 beats per minute (bpm).
Find the least value of \(n\) for Caleb’s heart rate to drop below 82 bpm.[3]
Caleb’s friend, Alex, also tracks his heart rate after training. His heart rate at the start of the first minute is 170 bpm and his heart rate can be modelled by the recurrence relation \(H_{n+1}=0.85H_n+10.5\), where \(H_n\) is his heart rate at the start of the \(n\)th minute after training.
Find the least value of \(n\) for Alex’s heart rate to drop below 90 bpm.[1]
Find his heart rate after he has rested long enough.[2]
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