Classify the sequence {a_n}={9,6,4,8/3} as arithmetic geometric or neither. What is the common difference or common ratio

Answer 1

The sequence {a_n} is neither an arithmetic nor a geometric sequence.

In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant.

For {a_n}={9,6,4,8/3}, the difference between consecutive terms is not constant:
6 - 9 = -3
4 - 6 = -2
8/3 - 4 = 4/3

Since the differences are not constant, the sequence is not arithmetic.

Similarly, the ratio between consecutive terms is not constant:
6/9 = 2/3
4/6 = 2/3
(8/3)/4 = 2/3

Since the ratios are not constant, the sequence is not geometric.

Therefore, the sequence {a_n}={9,6,4,8/3} is neither arithmetic nor geometric. There is no common difference or common ratio.