Question
Given the Recursive Formula: a1 = 5 a4 = an-1 + 2 a) What term is this formula solving for? b) What should be in place of the n-1?
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GPT-4o mini
Let's break down the recursive formula you've provided.
1. **Understanding the Recursive Formula**:
- \( a_1 = 5 \): This tells us that the first term of the sequence is 5.
- \( a_n = a_{n-1} + 2 \): This indicates that each term \( a_n \) is equal to the previous term \( a_{n-1} \) plus 2.
2. **a) What term is this formula solving for?**
- The formula is solving for the term \( a_n \), where \( n \) is greater than or equal to 2. It defines the sequence starting from the first term \( a_1 = 5 \) and describes how to find successive terms in the sequence by adding 2 to the previous term.
3. **b) What should be in place of \( n-1 \)?**
- In the recursive formula, \( n \) should represent the index of the current term. Therefore, \( n-1 \) represents the index of the previous term. For example:
- When \( n = 2 \), it refers to the second term \( a_2 \) and will use \( a_{1} \).
- When \( n = 3 \), it refers to the third term \( a_3 \) and will use \( a_{2} \).
- Thus, there is nothing that needs to be changed in the expression \( n-1 \); it is correctly indicating the previous term \( a_{n-1} \).
In summary:
a) The formula is solving for the term \( a_n \).
b) The notation \( n-1 \) correctly indicates the previous term \( a_{n-1} \) and does not need to be replaced.
1. **Understanding the Recursive Formula**:
- \( a_1 = 5 \): This tells us that the first term of the sequence is 5.
- \( a_n = a_{n-1} + 2 \): This indicates that each term \( a_n \) is equal to the previous term \( a_{n-1} \) plus 2.
2. **a) What term is this formula solving for?**
- The formula is solving for the term \( a_n \), where \( n \) is greater than or equal to 2. It defines the sequence starting from the first term \( a_1 = 5 \) and describes how to find successive terms in the sequence by adding 2 to the previous term.
3. **b) What should be in place of \( n-1 \)?**
- In the recursive formula, \( n \) should represent the index of the current term. Therefore, \( n-1 \) represents the index of the previous term. For example:
- When \( n = 2 \), it refers to the second term \( a_2 \) and will use \( a_{1} \).
- When \( n = 3 \), it refers to the third term \( a_3 \) and will use \( a_{2} \).
- Thus, there is nothing that needs to be changed in the expression \( n-1 \); it is correctly indicating the previous term \( a_{n-1} \).
In summary:
a) The formula is solving for the term \( a_n \).
b) The notation \( n-1 \) correctly indicates the previous term \( a_{n-1} \) and does not need to be replaced.
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