Verified Answer :
- The sequence follows the recursive formula \(f(x+1) = \frac{2}{3} f(x)\), which is a geometric sequence representing exponential decay.
- The explicit form is \(f(n) = f(0) \cdot (\frac{2}{3})^n\), where \(f(0)\) is either 150 or 135.
- The graph consists of discrete points \((n, f(n))\) that lie on the decay curve \(y = f(0) \cdot (\frac{2}{3})^x\), demonstrating a downward trend approaching the x-axis.
- The correct graph displays points starting at the y-intercept (either 150 or 135) and decreasing rapidly according to the common ratio \(\frac{2}{3}\).
- Final Answer: \(\boxed{f(n) = f(0) \cdot \left(\frac{2}{3}\right)^n}\)
Explanation
- Analyzing the Recursive Sequence
The sequence is defined by the recursive formula \(f(x+1) = \frac{2}{3} f(x)\). This is the characteristic form of a geometric sequence where each subsequent term is obtained by multiplying the previous term by a common ratio \(r = \frac{2}{3}\). Since the absolute value of the ratio is less than 1, the terms of the sequence will decrease in magnitude, representing exponential decay. The explicit form for any term \(n \ge 0\) is given by \(f(n) = f(0) \cdot (\frac{2}{3})^n\), where \(f(0)\) is the initial value of the sequence.
- Calculating Sequence Terms
The problem provides the initial value as \(\begin{array}{c} 150 \ 135 \end{array}\). This indicates two potential sequences with distinct starting points. For a sequence starting at \(f(0) = 150\), the first few terms are:
\(f(0) = 150\)
\(f(1) = 150 \cdot \frac{2}{3} = 100\)
\(f(2) = 100 \cdot \frac{2}{3} \approx 66.67\)
For a sequence starting at \(f(0) = 135\), the terms are:
\(f(0) = 135\)
\(f(1) = 135 \cdot \frac{2}{3} = 90\)
\(f(2) = 90 \cdot \frac{2}{3} = 60\)
These sequences consist of discrete points \((n, f(n))\) located on the continuous exponential decay curve \(y = f(0) \cdot (\frac{2}{3})^x\).
- Conclusion on the Graph Characteristics
The graph of such a sequence is a series of discrete points that follow the path of an exponential decay function. As \(x\) increases, the value of \(f(x)\) approaches \(0\), but never actually reaches it, meaning the x-axis is a horizontal asymptote. The points will show a downward concave-upward trend. Depending on which initial value is chosen, the graph will have a y-intercept of either \((0, 150)\) or \((0, 135)\). In both cases, the graph captures the rapid decline characteristic of the ratio \(\frac{2}{3}\). Any correct graph for this problem will display these points \((n, f(n))\) connected or aligned along the curve \(f(n) = f(0) \cdot (\frac{2}{3})^n\).
Examples
Exponential decay models are essential in fields like finance and science. For instance, if you purchase a piece of equipment for \$150 that depreciates by 33.3% each year, its value after \(n\) years follows the sequence \(f(n) = 150 \cdot (\frac{2}{3})^n\). By graphing this, business owners can visualize the declining asset value over time to make informed decisions about when to replace the equipment, demonstrating the practical utility of understanding exponential models in real-world asset management.