Which of the following sequences are not geometric?
Check all that apply.
A. -4, -2, -1, -0.5,-0.25, -0.125
B. 1, 4, 9, 16, 25, 36
C. 1, 1, 2, 3, 5, 8, 13, 21
D. 2, 10, 50, 250, 1250
Answer (2)
Check if the ratio between consecutive terms is constant for each sequence.
Sequence A has a constant ratio of 0.5, so it is geometric.
Sequence B does not have a constant ratio, so it is not geometric.
Sequence C does not have a constant ratio, so it is not geometric.
Sequence D has a constant ratio of 5, so it is geometric.
The sequences that are not geometric are B , C .
Explanation
Analyzing the Sequences Let's analyze each sequence to determine if it is geometric. A geometric sequence has a constant ratio between consecutive terms.
Sequence A Sequence A: -4, -2, -1, -0.5, -0.25, -0.125. The ratios between consecutive terms are: − 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 is constant (0.5), this sequence is geometric.
Sequence B Sequence B: 1, 4, 9, 16, 25, 36. The ratios between consecutive terms are: 1 4 = 4 , 4 9 = 2.25 , 9 16 ≈ 1.78 , 16 25 = 1.5625 , 25 36 = 1.44 . Since the ratios are not constant, this sequence is not geometric.
Sequence C Sequence C: 1, 1, 2, 3, 5, 8, 13, 21. The ratios between consecutive terms are: 1 1 = 1 , 1 2 = 2 , 2 3 = 1.5 , 3 5 ≈ 1.67 , 5 8 = 1.6 , 8 13 = 1.625 , 13 21 ≈ 1.62 . Since the ratios are not constant, this sequence is not geometric.
Sequence D Sequence D: 2, 10, 50, 250, 1250. The ratios between consecutive terms are: 2 10 = 5 , 10 50 = 5 , 50 250 = 5 , 250 1250 = 5 . Since the ratio is constant (5), this sequence is geometric.
Conclusion Therefore, the sequences that are not geometric are B and C.
Examples
Geometric sequences are useful in modeling various real-world phenomena, such as the growth of bacteria populations, the decay of radioactive substances, and the calculation of compound interest. For example, if you invest $1000 in an account that earns 5% interest compounded annually, the amounts at the end of each year form a geometric sequence: $1000, $1050, $1102.50, and so on. Understanding geometric sequences helps in predicting future values in these scenarios.
The sequences that are not geometric are B (1, 4, 9, 16, 25, 36) and C (1, 1, 2, 3, 5, 8, 13, 21). Sequence A has a constant ratio and is geometric, while Sequence D also maintains a constant ratio, making it geometric as well.
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