Question
Consider the sequence 100,50,25,12.5,6.25...
What is the recursive formula that represents the sequence for the term numbers 1,2,3,4
, and 5?
What is the recursive formula that represents the sequence for the term numbers 1,2,3,4
, and 5?
Answers
GPT-4o mini
The given sequence is 100, 50, 25, 12.5, 6.25...
To identify the recursive formula, we first observe how each term relates to the previous term.
1. **First Term (n=1)**: \( a_1 = 100 \)
2. **Second Term (n=2)**: \( a_2 = 50 = \frac{100}{2} = \frac{a_1}{2} \)
3. **Third Term (n=3)**: \( a_3 = 25 = \frac{50}{2} = \frac{a_2}{2} \)
4. **Fourth Term (n=4)**: \( a_4 = 12.5 = \frac{25}{2} = \frac{a_3}{2} \)
5. **Fifth Term (n=5)**: \( a_5 = 6.25 = \frac{12.5}{2} = \frac{a_4}{2} \)
From this pattern, we see that each term is obtained by dividing the previous term by 2.
Thus, the recursive formula can be expressed as follows:
- \( a_1 = 100 \)
- \( a_n = \frac{a_{n-1}}{2} \) for \( n \geq 2 \)
This recursive formula correctly describes the given sequence for the specified term numbers.
To identify the recursive formula, we first observe how each term relates to the previous term.
1. **First Term (n=1)**: \( a_1 = 100 \)
2. **Second Term (n=2)**: \( a_2 = 50 = \frac{100}{2} = \frac{a_1}{2} \)
3. **Third Term (n=3)**: \( a_3 = 25 = \frac{50}{2} = \frac{a_2}{2} \)
4. **Fourth Term (n=4)**: \( a_4 = 12.5 = \frac{25}{2} = \frac{a_3}{2} \)
5. **Fifth Term (n=5)**: \( a_5 = 6.25 = \frac{12.5}{2} = \frac{a_4}{2} \)
From this pattern, we see that each term is obtained by dividing the previous term by 2.
Thus, the recursive formula can be expressed as follows:
- \( a_1 = 100 \)
- \( a_n = \frac{a_{n-1}}{2} \) for \( n \geq 2 \)
This recursive formula correctly describes the given sequence for the specified term numbers.