You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.
Your classmate is incorrect. The sequence 2, 3, 5, 8, 12 is neither arithmetic nor geometric. To check if a sequence is arithmetic, we calculate the common difference between adjacent terms. For example, in the sequence 1, 4, 7, 10, the common difference is 3. However, the sequence 2, 3, 5, 8, 12 does not have a constant common difference.
To check if a sequence is geometric, we calculate the common ratio between adjacent terms. For example, in the sequence 1, 2, 4, 8, the common ratio is 2. However, the sequence 2, 3, 5, 8, 12 does not have a constant common ratio.
Therefore, the sequence 2, 3, 5, 8, 12 is neither arithmetic nor geometric. It is an example of a neither arithmetic nor geometric sequence.
Your classmate is incorrect. The sequence 2, 3, 5, 8, 12 is neither arithmetic nor geometric.
To determine if a sequence is arithmetic, we check if the difference between consecutive terms is constant. In this case, the differences between consecutive terms are 1, 2, 3, which are not constant. Therefore, the sequence is not arithmetic.
To determine if a sequence is geometric, we check if the ratio between consecutive terms is constant. If it is, then the sequence is geometric. In this case, the ratios between consecutive terms are 3/2, 5/3, 8/5, which are also not constant. Therefore, the sequence is not geometric.
Hence, the given sequence is neither arithmetic nor geometric.