MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series
Problem
To offset airborne pollutants, a city intends to install a sequence of filtration units. Each unit removes a fixed fraction r of the amount removed by the previous unit. The first unit is designed to remove a tons of pollutants in its first year of operation, the second removes ar tons in its first year, and so on. The city’s goal is to remove exactly 2,000 tons of pollutants in total over the lifetimes of all units combined.
(a) Write an equation that relates a and r to the city’s goal of removing 2,000 tons.
(b) City engineers have already committed to a ratio of r=0.85. Determine the value of a that satisfies the requirement in part (a).
(c) Using the values of a and r obtained in part (b), calculate the amount of pollutants removed by the first 10 units combined, and interpret the result relative to the city’s long-term goal.
(d) Explain why choosing a value of r greater than or equal to 1 would undermine the city’s objective, referencing the mathematical behavior of the series.