Use the rule for the 𝑛th term of a geometric sequence.  𝑡𝑛 = 𝑡1 𝑟(𝑛−1) where 𝑡𝑛 is 𝑛th term, 𝑡1 is first term, 𝑛 is term number and 𝑟 is common ratio

A theatre company has 8 members in its first production. It plans to expand the cast by a constant factor each year. If the cast grows geometrically with a common ratio of 1.5, find the predicted number of cast members in the 7th production. Show your working and give your answer to the nearest whole number.

A podcast downloads tracker records the total number of downloads for a new podcast series over its first few weeks after launch. The table below shows the cumulative total number of downloads at the end of selected weeks. The podcast creator earns $0.012 per download and will receive payment after 15 weeks. They predict that by week 15, they will have earned at least $500. Given that the total number of downloads is increasing geometrically, evaluate the reasonableness of this prediction.

A species of bacteria in a controlled laboratory culture is observed at regular time intervals. The table shows the population count (in thousands) at selected time points. Given that the population is growing exponentially as a geometric sequence, determine the predicted population at 35 hours. Show your working and round your final answer to the nearest thousand.

A telecommunications company releases a new software update to 12 devices on day 1. The update spreads to new devices each day, with the number of newly affected devices doubling every day. Which equation correctly represents the number of newly affected devices on day $n$?