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Answer:
To find the first term \(a\) and the common ratio \(r\) of a geometric sequence given that the 3rd term is 18 and the 6th term is 486, we can use the following properties of geometric sequences:
1. The \(n\)-th term of a geometric sequence is given by:
\[ a_n = a \cdot r^{(n-1)} \]
Given:
- The 3rd term (\(a_3\)) is 18, so:
\[ a \cdot r^2 = 18 \]
- The 6th term (\(a_6\)) is 486, so:
\[ a \cdot r^5 = 486 \]
Now, we can set up the equations:
\[ a \cdot r^2 = 18 \tag{1} \]
\[ a \cdot r^5 = 486 \tag{2} \]
To eliminate \(a\), divide equation (2) by equation (1):
\[ \frac{a \cdot r^5}{a \cdot r^2} = \frac{486}{18} \]
\[ r^3 = 27 \]
Solve for \(r\):
\[ r = \sqrt[3]{27} \]
\[ r = 3 \]
Now, substitute \(r = 3\) back into equation (1) to find \(a\):
\[ a \cdot 3^2 = 18 \]
\[ a \cdot 9 = 18 \]
\[ a = \frac{18}{9} \]
\[ a = 2 \]
Thus, the first term \(a\) is 2, and the common ratio \(r\) is 3.