Answer :
To determine which of the sequences is a geometric series, we need to check if the ratio between consecutive terms is constant for each series. Let's examine each sequence one by one.
1. Sequence: [tex]\(40, 50, 60, 70\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{50}{40} = 1.25 \)[/tex]
[tex]\( \frac{60}{50} = 1.2 \)[/tex]
[tex]\( \frac{70}{60} = 1.1667 \)[/tex]
- The ratios are not the same, so this is not a geometric series.
2. Sequence: [tex]\(40, 42, 44, 46\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{42}{40} = 1.05 \)[/tex]
[tex]\( \frac{44}{42} \approx 1.0476 \)[/tex]
[tex]\( \frac{46}{44} \approx 1.0455 \)[/tex]
- The ratios are not the same, so this is not a geometric series.
3. Sequence: [tex]\(40, 80, 40, 120\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{80}{40} = 2 \)[/tex]
[tex]\( \frac{40}{80} = 0.5 \)[/tex]
[tex]\( \frac{120}{40} = 3 \)[/tex]
- The ratios are not the same, so this is not a geometric series.
4. Sequence: [tex]\(40, 20, 10, 5\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{20}{40} = 0.5 \)[/tex]
[tex]\( \frac{10}{20} = 0.5 \)[/tex]
[tex]\( \frac{5}{10} = 0.5 \)[/tex]
- The ratios are the same (i.e., [tex]\(0.5\)[/tex]), hence this is a geometric series.
So, the geometric series among the given sequences is [tex]\(40, 20, 10, 5\)[/tex], making the correct answer:
[tex]\[ \boxed{4} \][/tex]
1. Sequence: [tex]\(40, 50, 60, 70\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{50}{40} = 1.25 \)[/tex]
[tex]\( \frac{60}{50} = 1.2 \)[/tex]
[tex]\( \frac{70}{60} = 1.1667 \)[/tex]
- The ratios are not the same, so this is not a geometric series.
2. Sequence: [tex]\(40, 42, 44, 46\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{42}{40} = 1.05 \)[/tex]
[tex]\( \frac{44}{42} \approx 1.0476 \)[/tex]
[tex]\( \frac{46}{44} \approx 1.0455 \)[/tex]
- The ratios are not the same, so this is not a geometric series.
3. Sequence: [tex]\(40, 80, 40, 120\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{80}{40} = 2 \)[/tex]
[tex]\( \frac{40}{80} = 0.5 \)[/tex]
[tex]\( \frac{120}{40} = 3 \)[/tex]
- The ratios are not the same, so this is not a geometric series.
4. Sequence: [tex]\(40, 20, 10, 5\)[/tex]
- Calculate the ratios between consecutive terms:
[tex]\( \frac{20}{40} = 0.5 \)[/tex]
[tex]\( \frac{10}{20} = 0.5 \)[/tex]
[tex]\( \frac{5}{10} = 0.5 \)[/tex]
- The ratios are the same (i.e., [tex]\(0.5\)[/tex]), hence this is a geometric series.
So, the geometric series among the given sequences is [tex]\(40, 20, 10, 5\)[/tex], making the correct answer:
[tex]\[ \boxed{4} \][/tex]
