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For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).
Full solution
Q. For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).
- Identify Sequence Type: To determine if the sequence is arithmetic or geometric, we need to look at the pattern of the terms given. An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms. Let's first check for an arithmetic sequence by finding the difference between consecutive terms.
- Calculate Difference for st & nd Terms: The first term is . The second term is . To find the difference between the second and the first term, we calculate .
- Calculate Difference for nd & rd Terms: Simplifying the difference, we get . The difference between the first two terms is .
- Confirm Arithmetic Sequence: Now let's find the difference between the third and the second term. The third term is . We calculate .
- Check Last Term Difference: Simplifying this difference, we get . The difference between the second and third terms is also .
- Check Last Term Difference: Simplifying this difference, we get . The difference between the second and third terms is also . Since the differences between consecutive terms are the same (), we can conclude that the sequence is an arithmetic sequence with a common difference of .
- Check Last Term Difference: Simplifying this difference, we get . The difference between the second and third terms is also . Since the differences between consecutive terms are the same (), we can conclude that the sequence is an arithmetic sequence with a common difference of . To confirm our conclusion, let's check the difference between the last given term, , and the term before it, . We calculate .
- Check Last Term Difference: Simplifying this difference, we get . The difference between the second and third terms is also . Since the differences between consecutive terms are the same (), we can conclude that the sequence is an arithmetic sequence with a common difference of . To confirm our conclusion, let's check the difference between the last given term, , and the term before it, . We calculate . Simplifying this difference, we get .
