Use recursion to generate a geometric sequence.

A geometric sequence is defined by the recursive rule $t_1 = 5$ and $t_n = 2 \times t_{n-1}$ for $n \geq 2$. (a) Find the first four terms of the sequence. [1] (b) Determine the common ratio. [1] (c) Calculate $t_8$, the eighth term of the sequence. [1]

A geometric sequence has first term $a = 18$ and common ratio $r = \frac{2}{3}$. Which recursive formula correctly generates the next term of this sequence?

A marine biologist monitors the decline in a local fish population. The table shows the estimated population, P_n, at the end of year n since monitoring began at the end of 2020. The population follows a geometric sequence with recursive rule P_(n+1) = r × P_n where P_1 = 18,400 and r is the common ratio. Use the recursive rule and the table to determine whether the fish population at the end of 2029 will be less than 15% of the population at the end of 2022.

A geometric sequence is defined by the recurrence relation \( t_1 = 5 \) and \( t_{n+1} = 3 \times t_n \) for \( n \geq 1 \). Determine the value of the fourth term.