1. **Stating the problem:** We have two sequences: (0, 6, 12, 18, 24, ...) and (1, 6, 36, 216, 1296, ...). We need to identify which is arithmetic, find its common difference, and think of other relationships with common differences. 2. **Definition of arithmetic sequence:** An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This difference is called the common difference $d$. 3. **Check the first sequence:** Calculate differences: $$6 - 0 = 6$$ $$12 - 6 = 6$$ $$18 - 12 = 6$$ $$24 - 18 = 6$$ All differences are equal to 6, so the first sequence is arithmetic with common difference $d = 6$. 4. **Check the second sequence:** Calculate differences: $$6 - 1 = 5$$ $$36 - 6 = 30$$ $$216 - 36 = 180$$ $$1296 - 216 = 1080$$ Differences are not constant, so the second sequence is not arithmetic. 5. **Other mathematical relationships with common differences:** Arithmetic sequences model situations with constant addition or subtraction, such as evenly spaced numbers, linear growth, or time intervals. For example, the sequence of even numbers (2, 4, 6, 8, ...) has common difference 2. **Final answers:** - The first sequence is arithmetic. - Its common difference is $6$. - Other relationships with common differences include linear functions and evenly spaced numbers.