Which of the following sequences are geometric?
Check all that apply.
A. $-4, -2, -1, -0.5, -0.25, -0.125$
B. $6, 18, 54, 162, 486$
C. $2, 5, 8, 11, 14, 17$
D. $2, 3, 5, 8, 13, 21$
Answer (2)
A geometric sequence has a constant ratio between consecutive terms.
Sequence A has a common ratio of 0.5 and is geometric.
Sequence B has a common ratio of 3 and is geometric.
Sequences C and D do not have a common ratio and are not geometric. Therefore, the answer is A and B. A , B
Explanation
Understanding Geometric Sequences We need to determine which of the given sequences are geometric. A geometric sequence is a sequence where the ratio between consecutive terms is constant. We will check each sequence to see if it satisfies this condition.
Analyzing Sequence A Sequence A: − 4 , − 2 , − 1 , − 0.5 , − 0.25 , − 0.125 To check if this is a geometric sequence, we calculate the ratio between consecutive terms: − 4 − 2 = 0.5 − 2 − 1 = 0.5 − 1 − 0.5 = 0.5 − 0.5 − 0.25 = 0.5 − 0.25 − 0.125 = 0.5 Since the ratio between consecutive terms is constant (0.5), sequence A is geometric.
Analyzing Sequence B Sequence B: 6 , 18 , 54 , 162 , 486 To check if this is a geometric sequence, we calculate the ratio between consecutive terms: 6 18 = 3 18 54 = 3 54 162 = 3 162 486 = 3 Since the ratio between consecutive terms is constant (3), sequence B is geometric.
Analyzing Sequence C Sequence C: 2 , 5 , 8 , 11 , 14 , 17 To check if this is a geometric sequence, we calculate the ratio between consecutive terms: 2 5 = 2.5 5 8 = 1.6 Since the ratio between consecutive terms is not constant, sequence C is not geometric.
Analyzing Sequence D Sequence D: 2 , 3 , 5 , 8 , 13 , 21 To check if this is a geometric sequence, we calculate the ratio between consecutive terms: 2 3 = 1.5 3 5 ≈ 1.667 Since the ratio between consecutive terms is not constant, sequence D is not geometric.
Conclusion Therefore, the geometric sequences are A and B.
Examples
Geometric sequences are useful in many areas of mathematics and in real life. For example, the amount of money in a bank account that earns a fixed interest rate increases geometrically each year. Similarly, the decay of a radioactive substance decreases geometrically over time. Understanding geometric sequences helps us model and predict these types of phenomena.
The geometric sequences from the provided options are A and B, as they have constant ratios of 0.5 and 3 respectively. Sequences C and D do not have a constant ratio and are not geometric. Therefore, the answer is A , B .
;
