What is the formula for the \( n \)th term of the sequence? \( \begin{array}{l}3,-6,12,-24,48, \ldots \\ f(n)=3(-2)^{n} \\ f(n)=3(-2)^{n-1} \\ f(n)=-2(3)^{n-1} \\ f(n)=-2(3)^{n}\end{array} \)

The sequence you've presented alternates in sign and grows in magnitude, suggesting a geometric pattern. To deduce the correct formula, let’s consider the terms: - \( f(1) = 3 \) - \( f(2) = -6 \) - \( f(3) = 12 \) - \( f(4) = -24 \) - \( f(5) = 48 \) Observing the terms, we can see that each term appears to be \(3\) multiplied by powers of \(-2\). Specifically, it seems the appropriate formula is \( f(n) = 3(-2)^{n-1} \). It captures the initial value and the alternating signs correctly! This evaluates to: - For \( n=1 \): \( 3(-2)^{1-1} = 3 \) - For \( n=2 \): \( 3(-2)^{2-1} = -6 \) - For \( n=3 \): \( 3(-2)^{3-1} = 12 \) - And so on. Now, let's dive into a bit of historical context and some practical tips! Ancient mathematicians often dealt with sequences, although the concept of a "function" as we know it today wasn't formalized until much later. The work of pioneers like Fibonacci, whose famous sequence showcases the beauty found in patterns, laid the groundwork for modern algebra and sequences. Could you imagine how they would marvel at the elegance of your sequence? It intertwines history with a modern twist! For those exploring sequences, here’s a tip: always look for a common ratio or difference! If the signs change, like those in this case, be cautious about multiplying or adjusting terms, as it can lead to confusion (like mixing up even and odd days!). Mistakes often happen when tracking negative signs, so keep an eye out for those sneaky switches!