- Convert mixed numbers to improper fractions: \(a_1 = \frac{5}{2}, a_2 = \frac{25}{2}, a_3 = \frac{125}{2}\).
- Find the common ratio by dividing consecutive terms: \(r = \frac{a_2}{a_1} = 5\).
- State the first term: \(a_1 = 2 \frac{1}{2}\).
- Write the recursive formula: \(a_n = a_{n-1} \cdot 5\).
- Final answer format: \(\boxed{a_1 = 2 \frac{1}{2} ; a_n = a_{n-1} \cdot 5}\).
Explanation
- Analyzing the Sequence
We are given a geometric sequence starting with \(2 \frac{1}{2}, 12 \frac{1}{2}, 62 \frac{1}{2}, 312 \frac{1}{2}, \ldots\). A geometric sequence is characterized by its first term \(a_1\) and a common ratio \(r\), where each subsequent term is obtained by multiplying the previous term by the common ratio. We need to find \(a_1\) and \(r\) to write the recursive formula \(a_n = a_{n-1} \cdot r\).
- Converting Terms to Improper Fractions
First, let's convert the given mixed numbers into improper fractions to simplify our calculations:
- \(a_1 = 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\)
- \(a_2 = 12 \frac{1}{2} = \frac{12 \times 2 + 1}{2} = \frac{25}{2}\)
- \(a_3 = 62 \frac{1}{2} = \frac{62 \times 2 + 1}{2} = \frac{125}{2}\)
- \(a_4 = 312 \frac{1}{2} = \frac{312 \times 2 + 1}{2} = \frac{625}{2}\)
- Calculating the Common Ratio
In a geometric sequence, the common ratio \(r\) is given by dividing any term by its preceding term:
- \(r = \frac{a_2}{a_1} = \frac{25/2}{5/2} = \frac{25}{5} = 5\)
- \(r = \frac{a_3}{a_2} = \frac{125/2}{25/2} = \frac{125}{25} = 5\)
- \(r = \frac{a_4}{a_3} = \frac{625/2}{125/2} = \frac{625}{125} = 5\)
Since the ratio is consistent, the common ratio is \(r = 5\).
- Formulating the Final Answer
The first term of the sequence is \(a_1 = 2 \frac{1}{2}\). The common ratio is \(r = 5\). The recursive formula for a geometric sequence is \(a_n = a_{n-1} \cdot r\). Substituting the ratio we found, the recursive formula is \(a_n = a_{n-1} \cdot 5\). Thus, the requested format is \(a_1 = 2 \frac{1}{2}\) and \(a_n = a_{n-1} \cdot 5\).
Examples
Geometric sequences appear in many real-world scenarios involving exponential growth, such as population increase, compound interest in finance, or the spread of a virus. Understanding the recursive formula helps in predicting future values based on current conditions, allowing for better planning and forecasting in various fields like economics, biology, and logistics. For example, if an investment grows by a constant percentage, you can use a geometric sequence to determine its value over time.