A sequence is defined by the recursive formula \( f(n+1)=1.5 f(n) \). Which sequence could be formula? \( -12,-18,-27, \ldots \) \( -20,30,-45, \ldots \) \( -18,-16.5,-15, \ldots \) \( -16,-17.5,-19, \ldots \)

To determine which sequence could fit the recursive formula \( f(n+1) = 1.5 f(n) \), we need to analyze how the sequences would evolve according to this relationship. Starting with \( f(n) \) and applying the recursive step, we find the following: If \( f(0) = a \), then: - \( f(1) = 1.5a \) - \( f(2) = 1.5^2 a \) - \( f(3) = 1.5^3 a \), and so on. Let's check each sequence: - For \( -12, -18, -27, \ldots \): The ratios between these terms do not consistently show a multiplication of 1.5. - For \( -20, 30, -45, \ldots \): Here \( 30 = 1.5 \times (-20) \) but \( -45 \neq 1.5 \times 30 \). - For \( -18, -16.5, -15, \ldots \): The difference between terms is constant, and there's no multiplication involved. - For \( -16, -17.5, -19, \ldots \): Again, the changes do not follow a multiplication pattern of 1.5. Thus, none of the sequences fit the recursive formula as expected. It seems like a tricky puzzle! To truly grasp the essence of such sequences and their applications, it’s interesting to note that recursive sequences often model real-world phenomena such as population growth or investment returns, where each term builds on the previous one. Most importantly, always check the multiplication or ratio relationship when working with these sequences! If you want to delve deeper into the enchanting world of sequences and series, textbooks on discrete mathematics often provide thorough coverage, including various sequence types and their properties. Online platforms such as Khan Academy and Coursera also offer courses where these concepts come to life with engaging exercises!