Determine whether each sequence is arithmetic, geometric, or neither. Write your answer before each number. If the sequence is arithmetic, give the common difference. If geometric, give the common ratio. Show your solution. If neither, write "No d/r" as your solution.
1. 6, 18, 54, 162...
2. 4, 10, 16, 22...
3. 1, 1, 2, 3, 5, 8...
4. 625, 125, 25, 5, ...
5. 1/2, 1/4, 1/6, 1/8...
6. 5, 8, 13, 21, 34...
7. -1296, 216, -36, 6...
8. 8, 2, 8, 7, 8, 7, 6...
9. -1/42, -1/35, -1/28, -1/21
10. 11, 2, -7, -16, ...
Answer (2)
To determine whether each sequence is arithmetic, geometric, or neither, we must check for a common difference or a common ratio.
The sequence 6, 18, 54, 162... This is a geometric sequence. Each term is obtained by multiplying the previous term by 3. Thus, the common ratio is r = 3 .
The sequence 4, 10, 16, 22... This is an arithmetic sequence. The difference between consecutive terms is 6. Thus, the common difference is d = 6 .
The sequence 1, 1, 2, 3, 5, 8... This sequence follows the Fibonacci sequence and is neither arithmetic nor geometric. No common difference or ratio exists, so we write "No d/r".
The sequence 625, 125, 25, 5, ... This is a geometric sequence. Each term is obtained by multiplying the previous term by 1/5 [ t e x ] . T h u s , t h eco mm o n r a t i o i s [ / t e x ] r = 5 1 .
The sequence 1/2, 1/4, 1/6, 1/8... This sequence appears to decrease by equal parts in the denominator but does not have a common difference or common ratio, so we write "No d/r".
The sequence 5, 8, 13, 21, 34... This is similar to the Fibonacci sequence, where each term is the sum of the two preceding ones. It is neither arithmetic nor geometric, so we write "No d/r".
The sequence -1296, 216, -36, 6... This is a geometric sequence. Each term is obtained by multiplying the previous term by − 6 1 . Thus, the common ratio is r = − 6 1 .
The sequence 8, 2, 8, 7, 8, 7, 6... There is no consistent pattern of addition or multiplication, making it neither arithmetic nor geometric, so we write "No d/r".
The sequence -1/42, -1/35, -1/28, -1/21... This is an arithmetic sequence. Each term has a constant difference of 42 1 between consecutive terms. Thus, the common difference is d = 42 1 .
The sequence 11, 2, -7, -16, ... This is an arithmetic sequence. The difference between each term is -9. Thus, the common difference is d = − 9 .
The sequences have been analyzed as follows: (1) geometric with a ratio of 3; (2) arithmetic with a difference of 6; (3) neither; (4) geometric with a ratio of 1/5; (5) neither; (6) neither; (7) geometric with a ratio of -1/6; (8) neither; (9) arithmetic with a difference of 1/42; (10) arithmetic with a difference of -9.
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