Infinite Geometric Series Worksheets Grade 12

1. Emma is analyzing a fractal pattern formed by a series of nested squares. The largest square has a side length of 20 cm. Each subsequent square is drawn inside the previous one such that its vertices touch the midpoints of the sides of the larger square, creating a pattern that continues infinitely inward. Determine the total area of all the squares in this infinite geometric series.
2. ∑(n=1 to ∞) 14(0.8)^(n-1) = ?
3. Tane is studying the population dynamics of an endangered tree fern species in a regenerating forest. He observes that each year, the number of new fronds produced by the population is exactly one-third of the number of fronds produced in the previous year. If the fern population produced 81 new fronds in the first year of his study, and this pattern of decline continues indefinitely, what is the total number of fronds that will ever be produced by this population from the first year onward?

…and 3 more problems

1. A pharmaceutical company is modeling the decay of a radioactive tracer in the human body. The amount of tracer remaining after each 24-hour period follows a geometric sequence where each term is 0.75 times the previous term. If the initial dose is 80 milligrams, what is the total amount of tracer that will eventually be present in the body over infinite time?
2. Mere is studying a fractal pattern formed by a series of nested circles. The largest circle has a radius of 12 cm. Each subsequent circle is drawn inside the previous one such that its diameter is exactly half the diameter of the larger circle, and this pattern continues infinitely inward. Determine the total area of all the circles in this infinite geometric series, giving your answer in terms of π.
3. Noah is a theoretical physicist modeling the behavior of a quantum harmonic oscillator. He discovers that the energy levels of the oscillator follow a specific pattern: the ground state energy is 48 femtojoules (fJ), and each successive excited state has an energy equal to 60% of the previous state's energy. If this infinite sequence of energy levels continues, what is the total energy (in fJ) of all possible states combined?

…and 2 more problems

1. Mason is studying the long-term accumulation of a new slow-release fertilizer in a garden plot. Each week, the soil absorbs 72 mg of the fertilizer from the latest application, but 30% of the existing fertilizer is consumed by plants or leached away by the end of each week. If Mason applies 72 mg of fertilizer every week indefinitely, what total amount of fertilizer (in mg) will be present in the soil in the long run?
2. Olivia is monitoring the spread of a beneficial bacteria in a bioreactor used for waste treatment. Each hour, the bacteria break down 80% of the remaining waste in the reactor, leaving 20% of the waste untouched. If the initial waste load is 500 kg, and the bacteria continue this pattern indefinitely, what is the total amount of waste (in kg) that the bacteria will eventually break down?
3. A pharmaceutical company is testing a new drug that is eliminated from the body at a rate such that 60% of the drug remains in the bloodstream at the end of each 8-hour dosing interval. If a patient takes a 250 mg dose every 8 hours, what is the maximum possible concentration of the drug in the bloodstream that will be approached over time?

…and 3 more problems