What is the explicit formula for this sequence?2, 6, 18, 54, 162, …A.an = 3(2)(n – 1)B.an = 2(3)nC.an = 2 + 3(n – 1)D.an = 2(3)(n – 1)

Break Down the Problem

1. Identify the pattern in the sequence: 2, 6, 18, 54, 162.
2. Determine the relationship between the terms and formulate the explicit formula.

Relevant Concepts

• This sequence can be analyzed to see if it follows a geometric progression. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant.

Analysis and Detail

1. Identify the first term: $a_1 = 2$

2. Calculate the ratio of successive terms:

   • $\frac{a_2}{a_1} = \frac{6}{2} = 3$
   • $\frac{a_3}{a_2} = \frac{18}{6} = 3$
   • $\frac{a_4}{a_3} = \frac{54}{18} = 3$
   • $\frac{a_5}{a_4} = \frac{162}{54} = 3$
   • The common ratio $r = 3$.

3. General formula for a geometric sequence: The $n$-th term of a geometric sequence can be expressed as: $a_n = a_1 \cdot r^{(n - 1)}$ Substituting the values we have: $a_n = 2 \cdot 3^{(n - 1)}$

Verify and Summarize

The derived formula $a_n = 2 \cdot 3^{(n - 1)}$ matches with the terms of the sequence:

• For $n = 1$: $2 \cdot 3^{(1-1)} = 2$
• For $n = 2$: $2 \cdot 3^{(2-1)} = 6$
• For $n = 3$: $2 \cdot 3^{(3-1)} = 18$
• For $n = 4$: $2 \cdot 3^{(4-1)} = 54$
• For $n = 5$: $2 \cdot 3^{(5-1)} = 162$

This confirms that our formula is correct.

Final Answer

The explicit formula for the sequence is: $a_n = 2(3)^{(n-1)}$

Thus, the correct answer is B. $a_n = 2(3)^{n-1}$.