To analyze the sequence given: \( a_n = 2, 6, 18, 54, 162 \), we can first observe how the terms relate to each other.
Looking at the ratios of consecutive terms:
- \( \frac{a_2}{a_1} = \frac{6}{2} = 3 \)
- \( \frac{a_3}{a_2} = \frac{18}{6} = 3 \)
- \( \frac{a_4}{a_3} = \frac{54}{18} = 3 \)
- \( \frac{a_5}{a_4} = \frac{162}{54} = 3 \)
We can see that each term is obtained by multiplying the previous term by \( 3 \). This indicates that the recursive relationship involves multiplying the previous term by \( 3 \).
Thus, we can write the recursive formula:
- \( a_1 = 2 \)
- \( a_n = 3a_{n-1} \)
Now, let’s evaluate the options:
Option #1: \( a_1 = 2; \ \ a_n = 3a_{n-1} \) (This is correct.)
Option #2: \( a_1 = 2; \ \ a_n = 3a_{n+1} \) (This is incorrect, as it does not follow the pattern.)
Option #3: \( a_1 = 2; \ \ a_n = 3a_{n-1} + 2 \) (This is incorrect, as it adds 2, which doesn't follow the pattern as seen in the original sequence.)
Therefore, the correct answer is:
Option #1: a1=2; an=3an−1