Question
Given the common difference and the first term of an arithmetic sequence, which formula, the recursive or explicit formula, would be more useful to determine a 500? Explain. Would your answer change if you knew the value of a 499? Explain.
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Part 1
Choose the correct answer below.
A.
The explicit formula would be more useful in both cases because it is the only way of finding terms with such great term numbers.
B.
The recursive formula would be more useful in both cases because it is straightforward and easy to use.
C.
If only the common difference and the first term are known, then the explicit formula would be more useful. It would not require the calculations of each previous term. If the value of a 499 were also known, then the recursive formula would be more useful. It would involve only a single addition.
D.
If only the common difference and the first term are known, then the recursive formula is more useful. It would not require the calculations of each previous term. If the value of a 499 were also known, then the explicit formula would be more useful. It would involve only a single addition.
Question content area bottom
Part 1
Choose the correct answer below.
A.
The explicit formula would be more useful in both cases because it is the only way of finding terms with such great term numbers.
B.
The recursive formula would be more useful in both cases because it is straightforward and easy to use.
C.
If only the common difference and the first term are known, then the explicit formula would be more useful. It would not require the calculations of each previous term. If the value of a 499 were also known, then the recursive formula would be more useful. It would involve only a single addition.
D.
If only the common difference and the first term are known, then the recursive formula is more useful. It would not require the calculations of each previous term. If the value of a 499 were also known, then the explicit formula would be more useful. It would involve only a single addition.
Answers
Answered by
GPT-4o mini
The correct answer is:
**C. If only the common difference and the first term are known, then the explicit formula would be more useful. It would not require the calculations of each previous term. If the value of a 499 were also known, then the recursive formula would be more useful. It would involve only a single addition.**
### Explanation:
1. **Finding \( a_{500} \) using the explicit formula**: In an arithmetic sequence, the explicit formula to find the nth term is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. If you want to find \( a_{500} \), you can simply plug \( n = 500 \) into this formula without needing to calculate prior terms.
2. **Using the recursive formula**: The recursive formula is defined as:
\[
a_n = a_{n-1} + d
\]
To find \( a_{500} \), you'd need to calculate \( a_{499} \), then \( a_{498} \), and so on, which can become cumbersome for large n. Therefore, the explicit formula is more advantageous when you want to find terms deep into the sequence without recalculating each preceding term.
3. **Knowing \( a_{499} \)**: If \( a_{499} \) is known, the recursive formula becomes more beneficial for finding \( a_{500} \) since you can simply compute:
\[
a_{500} = a_{499} + d
\]
This calculation requires only a single addition, making it straightforward and efficient if the previous term is already known.
Thus, option C correctly summarizes the advantages of the explicit and recursive formulas in the context given.
**C. If only the common difference and the first term are known, then the explicit formula would be more useful. It would not require the calculations of each previous term. If the value of a 499 were also known, then the recursive formula would be more useful. It would involve only a single addition.**
### Explanation:
1. **Finding \( a_{500} \) using the explicit formula**: In an arithmetic sequence, the explicit formula to find the nth term is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. If you want to find \( a_{500} \), you can simply plug \( n = 500 \) into this formula without needing to calculate prior terms.
2. **Using the recursive formula**: The recursive formula is defined as:
\[
a_n = a_{n-1} + d
\]
To find \( a_{500} \), you'd need to calculate \( a_{499} \), then \( a_{498} \), and so on, which can become cumbersome for large n. Therefore, the explicit formula is more advantageous when you want to find terms deep into the sequence without recalculating each preceding term.
3. **Knowing \( a_{499} \)**: If \( a_{499} \) is known, the recursive formula becomes more beneficial for finding \( a_{500} \) since you can simply compute:
\[
a_{500} = a_{499} + d
\]
This calculation requires only a single addition, making it straightforward and efficient if the previous term is already known.
Thus, option C correctly summarizes the advantages of the explicit and recursive formulas in the context given.