Question

arithmetic or geometric
2,4,12,48,240,....

Answer 1

To determine whether the sequence \(2, 4, 12, 48, 240, \ldots\) is arithmetic or geometric, we need to analyze the relationships between the consecutive terms.

1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant.
- Differences:
- \(4 - 2 = 2\)
- \(12 - 4 = 8\)
- \(48 - 12 = 36\)
- \(240 - 48 = 192\)

The differences (2, 8, 36, 192) are not constant, so this sequence is **not arithmetic**.

2. **Geometric Sequence**: In a geometric sequence, the ratio between consecutive terms is constant.
- Ratios:
- \( \frac{4}{2} = 2 \)
- \( \frac{12}{4} = 3 \)
- \( \frac{48}{12} = 4 \)
- \( \frac{240}{48} = 5 \)

The ratios (2, 3, 4, 5) are not constant, so this sequence is **not geometric** either.

Since the sequence is neither arithmetic nor geometric, we can further investigate the pattern of how the terms are generated.

Looking closely, we can observe that each term appears to be created by multiplying the previous term by a consecutive integer (i.e., \(2, 3, 4, 5\)):
- \(2 \times 2 = 4\)
- \(4 \times 3 = 12\)
- \(12 \times 4 = 48\)
- \(48 \times 5 = 240\)

It seems the \(n\)-th term can be expressed as:
\[
a_n = a_{n-1} \times n \quad \text{with } a_1 = 2
\]

Thus, the sequence follows a pattern where each term is obtained by multiplying the previous term by an increasing integer starting from 2.