Geometric Sequences Worksheets Grade 12

1. Is 9, 63, 441, 3087... geometric? Find common ratio r = ?
2. Isabella is analyzing the population decline of an endangered species in a wildlife reserve. The population was 3,125 individuals in 2010, and it has been decreasing by 20% each year. She wants to determine the population in 2015 and the year when the population will drop below 1,000 individuals. Show that this situation represents a geometric sequence, identify the common ratio, and write the explicit formula for the population after n years (where n = 0 corresponds to 2010). Then use your formula to answer her questions.
3. Liam is a financial analyst studying the depreciation of a specialized piece of manufacturing equipment. The equipment's value (in thousands of dollars) at the end of each year forms a geometric sequence. After the first year, the equipment is worth $405,000. After the third year, it is worth $245,000. Assuming the common ratio is positive and less than 1, determine the explicit formula for the value of the equipment V(n) after n years, where V(n) is in thousands of dollars.

…and 4 more problems

1. Olivia is an environmental scientist monitoring the population of a certain fish species in a lake. She observes that the population decreases by 20% each year due to environmental changes. The initial population is 10,000 fish. She wants to find the number of fish after n years. Is this a geometric sequence? If so, find the common ratio and write the explicit formula for the population after n years.
2. Tane is studying a fractal pattern made of circles. The largest circle has a radius of 27 cm. A second circle is drawn inside the first, tangent to it at a single point, with its center on a diameter of the larger circle. The radius of each subsequent circle is exactly one-third the radius of the previous circle. This process of drawing smaller tangent circles continues infinitely along the same diameter. What is the total length of all the diameters of every circle in this infinite pattern?
3. Is 7, 28, 112, 448... geometric? Find common ratio = ?

…and 3 more problems

1. Olivia is training for a long-distance cycling event. On her first day of training, she cycles 40 kilometers. Each subsequent day, she increases her distance by 20% of the previous day's distance. She plans to continue this pattern indefinitely. Determine the total distance Olivia will have cycled after an infinite number of days.
2. Aroha is a marine biologist studying the population of a rare species of fish in a protected lagoon. She observes that the population decreases by a constant ratio each year due to a natural predator. In the first year, she counts 729 fish. In the second year, she counts 243 fish. In the third year, she counts 81 fish. Aroha determines that this pattern continues to follow a geometric sequence. Write the explicit formula for the number of fish in the nth year, using the form aₙ = a₁ * r^(n-1).
3. Is 9, 54, 324, 1944... geometric? Find common ratio

…and 4 more problems