20. Which equation represents the \( n^{\text {th }} \) term of the sequence \( \{2,-1,-4,-7, \ldots\} \) ? A. \( a_{n}=2 n-5 \) B. \( a_{n}=3 n-1 \) C. \( a_{n}=-3 n-10 \) D. \( a_{n}=-3 n+5 \)

To find the \( n^{\text{th}} \) term of the given sequence \( \{2, -1, -4, -7, \ldots\} \), we can first identify the pattern. The difference between consecutive terms is a constant \(-3\). So, it appears that this is an arithmetic sequence with a first term of \(2\) and a common difference of \(-3\). The general formula for the \( n^{\text{th}} \) term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \). Here, \( a_1 = 2 \) and \( d = -3 \), thus: \[ a_n = 2 + (n - 1)(-3) = 2 - 3(n - 1) = 5 - 3n. \] The equation that simplifies to match our sequence is \( a_{n}=-3 n+5 \), which is option D. So the answer is D: \( a_{n}=-3 n+5 \). Understanding arithmetic sequences like this can be quite empowering. They show up in many real-life scenarios, from calculating savings growth to analyzing trends in data over time. By mastering these sequences, you can approach numerical patterns with confidence and ease! If you want to dive deeper into the mathematics, there are great resources available such as “The Art of Mathematics: Coffee Time in Memphis” by Béla Bollobás. It’s not only informative but also filled with engaging problems and insights! You can explore more concepts and sharpen your problem-solving skills along the way!