Sure, let's answer the question by finding a detailed, step-by-step solution.
We start with the given explicit formula for the geometric sequence:
[tex]\[ a_n = 12 \cdot (33)^{n-1} \][/tex]
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio. To convert the explicit formula into a recursive formula, we first need to identify the first term and the common ratio.
1. Identify the first term:
The first term, [tex]\( a_1 \)[/tex], is found by setting [tex]\( n = 1 \)[/tex]:
[tex]\[
a_1 = 12 \cdot (33)^{1-1} = 12 \cdot (33)^0 = 12 \cdot 1 = 12
\][/tex]
So, [tex]\( a_1 = 12 \)[/tex].
2. Identify the common ratio:
In the explicit formula, the common ratio [tex]\( r \)[/tex] is the base of the exponent with [tex]\( n-1 \)[/tex], which in this case is 33. Therefore:
[tex]\[
r = 33
\][/tex]
3. Write the recursive formula:
In a recursive formula for a geometric sequence, you express each term in terms of the previous term multiplied by the common ratio. The general form is:
[tex]\[
a_n = r \cdot a_{n-1}
\][/tex]
So, for our sequence, we substitute [tex]\( r \)[/tex] with 33:
[tex]\[
a_n = 33 \cdot a_{n-1}
\][/tex]
4. Combine with the initial term:
We already know that the first term [tex]\( a_1 = 12 \)[/tex]. Therefore, the recursive formula becomes:
[tex]\[
a_1 = 12
\][/tex]
[tex]\[
a_n = 33 \cdot a_{n-1} \quad \text{for} \quad n > 1
\][/tex]
Thus, the recursive formula matching the given explicit formula is:
[tex]\[ a_1 = 12, \quad a_n = 33 \cdot a_{n-1} \][/tex]
Therefore, the correct option is:
A. [tex]\( a_1 = 12, \quad a_n = 33 \cdot a_{n-1} \)[/tex]
