To derive an equation for the sequence 3, 6, 12, ..., let's first examine how the terms progress.
- Identifying the pattern:
- The first term \( a_1 = 3 \)
- The second term \( a_2 = 6 \)
- The third term \( a_3 = 12 \)
It appears that each term can be obtained from the previous term by multiplying. Specifically:
- \( 6 = 3 \times 2 \)
- \( 12 = 6 \times 2 \)
From this, we can see that each term is double the previous term. This suggests a relationship involving powers of 2.
- General formula:
The first term can be expressed as \( 3 = 3 \times 2^0 \).
- The second term can be expressed as \( 6 = 3 \times 2^1 \).
- The third term can be expressed as \( 12 = 3 \times 2^2 \).
Thus, the pattern here is \( a_n = 3 \times 2^{n-1} \), where \( n \) is the position in the sequence (with \( n = 1 \) for the first term).
- Final equation: So, the equation that describes the sequence is:
\[ a_n = 3 \times 2^{n-1} \]
This matches the given structure \( a_n = ()^n - 1 \) if we manipulate it slightly, but here we maintain the relationship as discovered.
Thus, the final equation for the sequence is:
\[ a_n = 3 \times 2^{n-1} \]