Question

a down n=2,6,18,54,162…. Determine which option is the correct subscript notation for the recursive formula

Answer 1

To identify the correct subscript notation for the recursive formula of the given sequence \( n = 2, 6, 18, 54, 162, \ldots \), we first need to look for a pattern in the sequence.

Let's analyze the given terms:

- \( n_1 = 2 \)
- \( n_2 = 6 \)
- \( n_3 = 18 \)
- \( n_4 = 54 \)
- \( n_5 = 162 \)

Now, let's determine how each term relates to the previous term:

- \( n_2 = 3 \times n_1 = 3 \times 2 = 6 \)
- \( n_3 = 3 \times n_2 = 3 \times 6 = 18 \)
- \( n_4 = 3 \times n_3 = 3 \times 18 = 54 \)
- \( n_5 = 3 \times n_4 = 3 \times 54 = 162 \)

From this, we can see a pattern: Each term is multiplied by 3 to get the next term in the sequence.

Thus, we can express the recursive formula as:

\[
n_k = 3 \times n_{k-1} \quad \text{for } k \geq 2
\]

We also need to establish the initial condition:

\[
n_1 = 2
\]

Putting it all together, the recursive formula in subscript notation is:

\[
n_k = 3 \times n_{k-1}, \quad n_1 = 2 \quad \text{for } k \geq 2
\]

If you need to identify which among the given options matches this notation, look for the recursive relationship along with the initial condition specified above.