In a geometric sequence, the \( n \)-th term can be represented by the formula:
\[ a_n = a_1 \cdot r^{n-1} \]
where \( a_1 \) is the first term and \( r \) is the common ratio.
Given that the first term \( a_1 = 3 \) and the second term \( a_2 = \frac{3}{2^2} = \frac{3}{4} \), we can find the common ratio \( r \):
\[ a_2 = a_1 \cdot r \implies \frac{3}{4} = 3 \cdot r \] \[ r = \frac{\frac{3}{4}}{3} = \frac{1}{4} \]
Now that we have the common ratio \( r = \frac{1}{4} \), we can find the expression for the hundredth term \( a_{100} \):
\[ a_{100} = a_1 \cdot r^{100-1} = 3 \cdot \left( \frac{1}{4} \right)^{99} \]
This can also be expressed as:
\[ a_{100} = 3 \cdot \frac{1^{99}}{4^{99}} = 3 \cdot \frac{1}{4^{99}} = \frac{3}{4^{99}} = \frac{3}{(2^2)^{99}} = \frac{3}{2^{198}} \]
Now rewriting the term we found:
\[ a_{100} = \frac{3}{2^{198}} = \frac{3}{2^{198}} = \frac{3}{2^{198}} \cdot 1 = \frac{3}{2^{198}} \]
To match the answer options:
Since \( a_{100} = \frac{3}{2^{198}} \), and in the provided options, the equivalent term is:
- Option C: \(\frac{3}{2^{198}}\)
Therefore, the correct choice is:
C. \(\frac{3}{2^{198}}\)
1 answer