- The sequence \(-4, -6, -9, -\frac{27}{2}, -\frac{81}{4}, \dots\) is a geometric sequence with first term \(a = -4\) and common ratio \(r = \frac{3}{2}\).
- The explicit formula for this sequence is \(f(n) = -4 \cdot \left(\frac{3}{2}\right)^{(n-1)}\).
- The recursive definition is \(f(1) = -4\) and \(f(n) = \frac{3}{2} \cdot f(n-1)\) for \(n \geq 2\).
- The correct functions are:
- \(f(n) = -4 \cdot \left(\frac{3}{2}\right)^{(n-1)}\)
- \(f(1) = -4, f(n) = \frac{3}{2} \cdot f(n-1)\)
Explanation
- Analysis of the Sequence
The sequence provided is \(-4, -6, -9, -\frac{27}{2}, -\frac{81}{4}, \dots\). To identify the defining functions for this sequence, we first analyze its nature by finding the first term and the common ratio.
The first term is given by \(a_1 = -4\).
To find the common ratio \(r\), we divide any term by its preceding term:
\[r = \frac{a_2}{a_1} = \frac{-6}{-4} = \frac{3}{2}\]
We verify this with other consecutive terms:
\[\frac{a_3}{a_2} = \frac{-9}{-6} = \frac{3}{2}\]
\[\frac{a_4}{a_3} = \frac{-27/2}{-9} = \frac{27/2}{9} = \frac{3}{2}\]
Since the ratio is constant, it is indeed a geometric sequence with first term \(a_1 = -4\) and common ratio \(r = \frac{3}{2}\).
- Evaluating the Options
Now we evaluate the given functional forms:
- Explicit formula: A geometric sequence with first term \(a\) and common ratio \(r\) is given by \(f(n) = a \cdot r^{(n-1)}\). Substituting our values, we get \(f(n) = -4 \cdot \left(\frac{3}{2}\right)^{(n-1)}\).
- Recursive formula: A geometric sequence is defined by its first term and the relation \(f(n) = r \cdot f(n-1)\). With our values, this is \(f(1) = -4\) and \(f(n) = \frac{3}{2} \cdot f(n-1)\) for \(n \geq 2\).
Evaluating the provided options:
- \(f(n) = -4 \cdot \left(\frac{3}{2}\right)^{(n-1)}\): This matches our derived explicit formula.
- \(f(n) = 4 \cdot \left(-\frac{2}{3}\right)^{(n-1)}\): For \(n=1\), \(f(1) = 4
eq -4\). Incorrect.
- \(f(1) = 4, f(n) = -\frac{2}{3} \cdot f(n-1)\): First term is incorrect. Incorrect.
- \(f(1) = -4, f(n) = \frac{3}{2} \cdot f(n-1)\): This matches our derived recursive formula.
- Conclusion
The functions that correctly define the sequence are:
- The explicit formula: \(f(n) = -4 \cdot \left(\frac{3}{2}\right)^{(n-1)}\)
- The recursive formula: \(f(1) = -4\), \(f(n) = \frac{3}{2} \cdot f(n-1)\) for \(n \geq 2\).
Examples
Geometric sequences are frequently used to model growth or decay processes in real-world scenarios. For example, if a company's debt decreases by a consistent percentage each quarter, or if the value of an investment grows at a fixed interest rate, these values can be modeled using a geometric sequence. By identifying the correct explicit or recursive formula, you can predict future values in the sequence without having to calculate every intermediate step, which is an essential tool in financial planning and scientific modeling.