The recursive definition for the sequence 2, 6, 18, 54 is *an = 3 an-1**, where an represents the nth term of the sequence, and an-1 is the term preceding it.
Understanding the Recursive Definition
A recursive definition describes a sequence by relating each term to the previous terms. In this case, we find the next number in the sequence by multiplying the previous one by 3.
- Initial Term: The sequence starts with 2. This is our base case, a1 = 2.
- Recursive Step: To get the second term, we multiply the first term by 3: 2 * 3 = 6. This gives a2 = 6.
- Subsequent Terms: Similarly, to get the third term, we multiply the second term by 3: 6 * 3 = 18. This gives a3 = 18, and so on.
Breakdown
| Term Number (n) | Term (an) | Calculation |
|---|---|---|
| 1 | 2 | Given |
| 2 | 6 | 2 * 3 = 6 |
| 3 | 18 | 6 * 3 = 18 |
| 4 | 54 | 18 * 3 = 54 |
This pattern clearly demonstrates how each term is generated recursively from the previous one, according to the rule an = 3 an-1*, as stated in the reference: “the recursive rule for the sequence 2, 6, 18, 54, . . . is. an = 3 an − 1”.
