Makakuha ng mga sagot mula sa komunidad at mga eksperto sa IDNStudy.com. Magtanong ng anumang bagay at makatanggap ng kumpleto at eksaktong sagot mula sa aming komunidad ng mga propesyonal.
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.