2026 ASRJC P1 Q8

For this question, all answers for distance are to be expressed in kilometres.

Madilyn and Nancy are participating in a 60-day charity running challenge. The amount donated by their sponsors depends on the total distance each participant runs during the challenge.

Madilyn runs on alternate days, starting on Day 1. On Day 1, she runs 2 km. On each subsequent running day, she runs 100 m more than she did on her previous run. Her running distances are therefore 2 km on Day 1, 2.1 km on Day 3, 2.2 km on Day 5, and so on.

Nancy runs every day. On Day 1, she runs 2 km. On Day 2, she runs 200 m more than she did on Day 1. On each subsequent day, the additional distance she runs is 90% of the additional distance she ran on the previous day. Her daily running distances are therefore 2 km, 2.2 km, 2.38 km, and so on.

At the end of the 60-day challenge,

1. find the total distance run by Madilyn.[1]
2. Show that the daily distance, \(d_n\), run by Nancy on Day \(n\) is in the form \(4-k(0.9)^{n-1}\), where \(k\) is a real constant. Hence find the total distance covered by Nancy in 60 days. Leave your answer correct to 2 decimal places.[4]
3. Show that Nancy's daily running distance converges to a limiting value. Hence, determine the limit of her daily running distance.[2]
4. Determine the day on which Madilyn will first run more daily distance than Nancy.[1]