To analyze the geometric sequence given: \(-4, -12, -36\), we can identify the first term and the common ratio.
1) Create an equation for the situation
The first term (\(a\)) of the sequence is:
\[ a = -4 \]
To find the common ratio (\(r\)), we can divide the second term by the first term:
\[ r = \frac{-12}{-4} = 3 \]
Now we can express the \(n\)-th term of a geometric sequence as:
\[ a_n = a \cdot r^{(n-1)} \]
Substituting \(a = -4\) and \(r = 3\):
\[ a_n = -4 \cdot 3^{(n-1)} \]
2) Find the 12th term of the sequence
To find the 12th term (\(a_{12}\)), we can substitute \(n = 12\) into the equation:
\[ a_{12} = -4 \cdot 3^{(12-1)} = -4 \cdot 3^{11} \]
Calculating \(3^{11}\):
\[ 3^{11} = 177147 \]
Now we can calculate \(a_{12}\):
\[ a_{12} = -4 \cdot 177147 = -708588 \]
Conclusion
- The equation for the geometric sequence is:
\[ a_n = -4 \cdot 3^{(n-1)} \]
- The 12th term of the sequence is:
\[ a_{12} = -708588 \]
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