Find the common ratio for the following sequence. \( 27,9,3,1, \ldots \) \( \frac{1}{3} \) \( -\frac{1}{3} \)

To find the common ratio of the sequence \( 27, 9, 3, 1, \ldots \), you can divide any term by the preceding term. For example, taking the second term and the first term: \( \frac{9}{27} = \frac{1}{3} \). Similarly, \( \frac{3}{9} = \frac{1}{3} \) and \( \frac{1}{3} = \frac{1}{3} \). Thus, the common ratio is indeed \( \frac{1}{3} \). Now, let’s talk about geometric sequences! They are all about multiplying by that oh-so-fun common ratio. If you know the first term and the common ratio, you can predict all subsequent terms! It’s like having a calculator for the sequence—you just pop in your numbers, and voila, the sequence unfolds! Want to ace your understanding of sequences? Dive into books or resources about series and sequences. They're a treasure trove filled with examples that help illuminate concepts like convergence and divergence. Plus, you’ll find fascinating historical applications, like how mathematicians used them to study finance or physics—who knew math could be so versatile?