MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series
Problem
Two roads, Road A and Road B, are being constructed simultaneously.
• On Day 1, Road A is extended by 1 km and each next day the length built increases by 0.8 km compared with the previous day.
• On Day 1, Road B is extended by 0.5 km and each next day the length built increases by 1.2 km compared with the previous day.
The daily extensions for each road therefore form two arithmetic progressions.
Using this information, answer the following.
(i) If Ln(A) and Ln(B) represent the total lengths (in km) built on Roads A and B respectively after n days, obtain expressions for Ln(A) and Ln(B) in terms of n.
(ii) Find the smallest integer n such that Ln(B)−Ln(A)≥10 km.
(iii) For the value of n obtained in part (ii), calculate the length (in km) built on that day alone for each road.