Question

An incomplete table of sequences and common ratios is shown.(4 points)
Complete the table.
Sequence Common Ratio
3, 6, 12, 24, 48,.... ?
15, 9, 275, 8125, 243125... ?

Answer 1

To find the common ratios for the given sequences, we'll analyze each one:

1. For the sequence \(3, 6, 12, 24, 48, \ldots\):
- The common ratio \(r\) can be found by dividing any term by the previous term.
- \( r = \frac{6}{3} = 2 \)
- \( r = \frac{12}{6} = 2 \)
- \( r = \frac{24}{12} = 2 \)
- \( r = \frac{48}{24} = 2 \)
- Thus, the common ratio for this sequence is **2**.

2. For the sequence \(15, 9, 275, 8125, 243125, \ldots\):
- Let's calculate the ratios between consecutive terms:
- \( r = \frac{9}{15} = \frac{3}{5} \) (the first common ratio)
- \( r = \frac{275}{9} \approx 30.56 \)
- \( r = \frac{8125}{275} = 29.545 \)
- \( r = \frac{243125}{8125} = 30 \)
- This sequence does not have a constant common ratio, indicating it is not a geometric sequence.

Therefore, the completed table is:

| Sequence | Common Ratio |
|----------------------------------|--------------|
| 3, 6, 12, 24, 48, ... | 2 |
| 15, 9, 275, 8125, 243125, ... | No constant ratio |