Question
Use the input-output table for an arithmetic sequence. Input 1 2 3 4 Output 35.7 40.6 45.5 50.4 What is the recursive formula? What is the explicit formula? (1 point) Responses Recursive: a1=35.7; an=a(n−1)+5.1 Explicit: an=35.7+(n−1)(5.1) Recursive: A sub 1 is equal to 35 point 7 ; A sub n is equal to A sub open paren n minus 1 close paren plus 5 point 1 Explicit: A sub n is equal to 35 point 7 plus open paren n minus 1 close paren times 5 point 1 Recursive: a1=35.7; an=a(n−1)−4.9 Explicit: an=35.7+(n−1)(−4.9) Recursive: A sub 1 is equal to 35 point 7 ; A sub n is equal to A sub open paren n minus 1 close paren minus 4 point 9 Explicit: A sub n is equal to 35 point 7 plus open paren n minus 1 close paren times negative 4 point 9 Recursive: a1=35.7; an=a(n−1)+4.9 Explicit: an=35.7+(n−1)(4.9) Recursive: A sub 1 is equal to 35 point 7 ; A sub n is equal to A sub open paren n minus 1 close paren plus 4 point 9 Explicit: A sub n is equal to 35 point 7 plus open paren n minus 1 close paren times 4 point 9 Recursive: a1=1; an=a(n−1)+4.9 Explicit: an=1+(n−1)(4.9)
Answers
GPT-4o mini
To determine the recursive and explicit formulas for the given arithmetic sequence with the provided outputs, we first analyze the output values:
Given Outputs:
- When input is 1, output is 35.7
- When input is 2, output is 40.6
- When input is 3, output is 45.5
- When input is 4, output is 50.4
Looking at the differences between consecutive outputs:
- 40.6 - 35.7 = 4.9
- 45.5 - 40.6 = 4.9
- 50.4 - 45.5 = 4.9
The common difference \( d \) is 4.9.
### Recursive Formula
The recursive formula for an arithmetic sequence is given by:
- Base Case: \( a_1 = \) first term
- Recursive Step: \( a_n = a_{n-1} + d \)
Using the values:
- \( a_1 = 35.7 \)
- \( d = 4.9 \)
So the recursive formula is:
- **Recursive**: \( a_1 = 35.7 \); \( a_n = a_{n-1} + 4.9 \)
### Explicit Formula
The explicit formula for an arithmetic sequence is given by:
- \( a_n = a_1 + (n - 1)d \)
Using the values:
- \( a_1 = 35.7 \)
- \( d = 4.9 \)
So the explicit formula is:
- **Explicit**: \( a_n = 35.7 + (n - 1)(4.9) \)
### Final Answers
- **Recursive**: \( a_1 = 35.7 \); \( a_n = a_{n-1} + 4.9 \)
- **Explicit**: \( a_n = 35.7 + (n - 1)(4.9) \)
Given Outputs:
- When input is 1, output is 35.7
- When input is 2, output is 40.6
- When input is 3, output is 45.5
- When input is 4, output is 50.4
Looking at the differences between consecutive outputs:
- 40.6 - 35.7 = 4.9
- 45.5 - 40.6 = 4.9
- 50.4 - 45.5 = 4.9
The common difference \( d \) is 4.9.
### Recursive Formula
The recursive formula for an arithmetic sequence is given by:
- Base Case: \( a_1 = \) first term
- Recursive Step: \( a_n = a_{n-1} + d \)
Using the values:
- \( a_1 = 35.7 \)
- \( d = 4.9 \)
So the recursive formula is:
- **Recursive**: \( a_1 = 35.7 \); \( a_n = a_{n-1} + 4.9 \)
### Explicit Formula
The explicit formula for an arithmetic sequence is given by:
- \( a_n = a_1 + (n - 1)d \)
Using the values:
- \( a_1 = 35.7 \)
- \( d = 4.9 \)
So the explicit formula is:
- **Explicit**: \( a_n = 35.7 + (n - 1)(4.9) \)
### Final Answers
- **Recursive**: \( a_1 = 35.7 \); \( a_n = a_{n-1} + 4.9 \)
- **Explicit**: \( a_n = 35.7 + (n - 1)(4.9) \)