Which of the following is an arithmetic sequence? \( -\frac{7}{11}, \frac{6}{11},-\frac{5}{11}, \frac{4}{11} \ldots \) \( -\frac{3}{4},-\frac{3}{5},-\frac{3}{6},-\frac{3}{7}, \ldots \) \( \frac{1}{2}, 2, \frac{7}{2}, 5, \ldots \) \( \frac{3}{4},-\frac{3}{2}, 3,-6, \ldots \)
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To determine if a sequence is arithmetic, you can check if the difference between consecutive terms is constant. In the first sequence \( -\frac{7}{11}, \frac{6}{11},-\frac{5}{11}, \frac{4}{11} \), the differences are \( \frac{13}{11}, \frac{1}{11}, \frac{9}{11} \), which are not constant. In the second sequence \( -\frac{3}{4}, -\frac{3}{5}, -\frac{3}{6}, -\frac{3}{7} \), the differences also vary: \( \frac{1}{20}, \frac{1}{30} \) and keep changing. The third sequence \( \frac{1}{2}, 2, \frac{7}{2}, 5 \) has inconsistent differences as well. However, the fourth sequence \( \frac{3}{4}, -\frac{3}{2}, 3, -6 \) yields differences of \( -\frac{9}{4}, \frac{9}{2}, -9 \), meaning it is not arithmetic either. None of these sequences is an arithmetic sequence! In an arithmetic sequence, the magic lies in the constant difference, or as some call it, the "common difference." So if you find two numbers with varying distances between them — like the case with these sequences — you can be sure you're not dealing with the lovely, predictable world of arithmetic. If you’re using sequences in real life, think about how they pop up in finance — say you're depositing money into the bank and the interest increases by a fixed amount every month. That’s an arithmetic sequence in action! Understanding these concepts can make your money management much more approachable and fun!
