Question
The 3rd term of an arithmetic sequence is 32
. The common difference is 12
. Complete the statement.(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The explicit formula for the sequence is Response area and the recursive formula for the sequence is Response area.
Question 2
5. The first time Mya competed in a 500-meter freestyle race, her time was 15 minutes flat. In each subsequent race, Mya's time was reduced by 10 seconds.
Complete the statement.
(4 points)
The context is best represented by
and the domain is
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. The common difference is 12
. Complete the statement.(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The explicit formula for the sequence is Response area and the recursive formula for the sequence is Response area.
Question 2
5. The first time Mya competed in a 500-meter freestyle race, her time was 15 minutes flat. In each subsequent race, Mya's time was reduced by 10 seconds.
Complete the statement.
(4 points)
The context is best represented by
and the domain is
Skip to navigation
Answers
GPT-4o mini
**Question 1:**
The explicit formula for the arithmetic sequence can be calculated using the nth term formula for arithmetic sequences, which is:
\[ a_n = a_1 + (n - 1)d \]
Where:
- \( a_n \) is the nth term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
We know that the 3rd term \( a_3 = 32 \) and the common difference \( d = 12 \).
To find \( a_1 \):
\[ a_3 = a_1 + (3 - 1)(12) \]
\[ 32 = a_1 + 24 \]
\[ a_1 = 32 - 24 \]
\[ a_1 = 8 \]
Now, we have \( a_1 = 8 \) and \( d = 12 \).
The explicit formula for the sequence is:
\[ a_n = 8 + (n - 1) \cdot 12 \]
The recursive formula for the sequence is:
\[ a_n = a_{n-1} + 12 \text{ with } a_1 = 8 \]
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**Question 2:**
The context of Mya's racing times can be represented mathematically as a sequence, where her time decreases by 10 seconds for each subsequent race. The initial time (first term) is 15 minutes, which can also be converted to seconds for ease of calculation:
15 minutes = 900 seconds.
The time in each race can then be described recursively as:
- **The context is best represented by a decreasing arithmetic sequence.**
- **The domain is the set of natural numbers (n = 1, 2, 3, ...).**
In summary:
- The context is best represented by: **a decreasing arithmetic sequence.**
- The domain is: **the set of natural numbers (n = 1, 2, 3, ...)**.
The explicit formula for the arithmetic sequence can be calculated using the nth term formula for arithmetic sequences, which is:
\[ a_n = a_1 + (n - 1)d \]
Where:
- \( a_n \) is the nth term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
We know that the 3rd term \( a_3 = 32 \) and the common difference \( d = 12 \).
To find \( a_1 \):
\[ a_3 = a_1 + (3 - 1)(12) \]
\[ 32 = a_1 + 24 \]
\[ a_1 = 32 - 24 \]
\[ a_1 = 8 \]
Now, we have \( a_1 = 8 \) and \( d = 12 \).
The explicit formula for the sequence is:
\[ a_n = 8 + (n - 1) \cdot 12 \]
The recursive formula for the sequence is:
\[ a_n = a_{n-1} + 12 \text{ with } a_1 = 8 \]
---
**Question 2:**
The context of Mya's racing times can be represented mathematically as a sequence, where her time decreases by 10 seconds for each subsequent race. The initial time (first term) is 15 minutes, which can also be converted to seconds for ease of calculation:
15 minutes = 900 seconds.
The time in each race can then be described recursively as:
- **The context is best represented by a decreasing arithmetic sequence.**
- **The domain is the set of natural numbers (n = 1, 2, 3, ...).**
In summary:
- The context is best represented by: **a decreasing arithmetic sequence.**
- The domain is: **the set of natural numbers (n = 1, 2, 3, ...)**.