Which of the following are geometric series?
[tex]$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}$[/tex]
[tex]$2+5+8+11+14+17$[/tex]
[tex]$1+1+2+3+5+8+13+21$[/tex]
[tex]$-256+64-16+4-1$[/tex]
Answer (2)
Calculate the ratios between consecutive terms for each series.
Series 1 has a constant ratio of 2 1 , so it is geometric.
Series 2 and 3 do not have constant ratios, so they are not geometric.
Series 4 has a constant ratio of − 4 1 , so it is geometric.
The geometric series are 2 1 + 4 1 + 8 1 + 16 1 + 32 1 and − 256 + 64 − 16 + 4 − 1 .
Explanation
Understanding Geometric Series We are given four series and we need to determine which of them are geometric series. A geometric series is a sequence where the ratio between consecutive terms is constant. We will calculate the ratios between consecutive terms for each series.
Analyzing Series 1 For the first series, 2 1 + 4 1 + 8 1 + 16 1 + 32 1 , the ratios between consecutive terms are: 1/2 1/4 = 2 1 1/4 1/8 = 2 1 1/8 1/16 = 2 1 1/16 1/32 = 2 1 Since the ratio is constant and equal to 2 1 , the first series is geometric.
Analyzing Series 2 For the second series, 2 + 5 + 8 + 11 + 14 + 17 , the ratios between consecutive terms are: 2 5 = 2.5 5 8 = 1.6 8 11 = 1.375 11 14 ≈ 1.27 14 17 ≈ 1.21 Since the ratios are not constant, the second series is not geometric.
Analyzing Series 3 For the third series, 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 ≈ 1.615 Since the ratios are not constant, the third series is not geometric.
Analyzing Series 4 For the fourth series, − 256 + 64 − 16 + 4 − 1 , the ratios between consecutive terms are: − 256 64 = − 4 1 64 − 16 = − 4 1 − 16 4 = − 4 1 4 − 1 = − 4 1 Since the ratio is constant and equal to − 4 1 , the fourth series is geometric.
Conclusion Therefore, the geometric series are: 2 1 + 4 1 + 8 1 + 16 1 + 32 1 and − 256 + 64 − 16 + 4 − 1 .
Examples
Geometric series are useful in many areas of mathematics and physics. For example, they can be used to model the decay of radioactive substances, the growth of populations, and the behavior of financial markets. Imagine you invest $1000 in a savings account that earns 5% interest per year, compounded annually. The amounts you have at the end of each year form a geometric sequence: $1000, $1050, $1102.50, and so on. Understanding geometric series helps you predict how your investment will grow over time.
The geometric series are 2 1 + 4 1 + 8 1 + 16 1 + 32 1 and − 256 + 64 − 16 + 4 − 1 . The second and third series are not geometric because their ratios between consecutive terms are not constant. Therefore, the correct answers are the first and the fourth series.
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