Let's analyze each function based on the information given.
Martha designs 2 rings in the first hour and every additional hour she designs 3 more rings. We need to find which functions correctly describe the number of rings, [tex]\( r(n) \)[/tex], she designs in [tex]\( n \)[/tex] hours.
### 1. [tex]\( r(1)=3 ; r(n)=r(n-1)+2 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- For [tex]\( n = 1 \)[/tex]: [tex]\( r(1) = 3 \)[/tex] [This does not match the given initial condition [tex]\( r(1)=2 \)[/tex]]
- Therefore, this function does not satisfy the criteria.
### 2. [tex]\( r(1)=2 ; r(n)=r(n-1)+3 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- For [tex]\( n = 1 \)[/tex]: [tex]\( r(1) = 2 \)[/tex] [matches initial condition]
- For [tex]\( n = 2 \)[/tex]: [tex]\( r(2) = r(1) + 3 = 2 + 3 = 5 \)[/tex]
- For [tex]\( n = 3 \)[/tex]: [tex]\( r(3) = r(2) + 3 = 5 + 3 = 8 \)[/tex]
- This function correctly describes the number of rings designed each hour after the first.
### 3. [tex]\( r(n)=3n-1 \)[/tex]
- For [tex]\( n = 1 \)[/tex]: [tex]\( r(1) = 3(1) - 1 = 2 \)[/tex] [matches initial condition]
- For [tex]\( n = 2 \)[/tex]: [tex]\( r(2) = 3(2) - 1 = 5 \)[/tex]
- For [tex]\( n = 3 \)[/tex]: [tex]\( r(3) = 3(3) - 1 = 8 \)[/tex]
- This function also correctly describes the number of rings designed over time.
### 4. [tex]\( r(n)=3n+2 \)[/tex]
- For [tex]\( n = 1 \)[/tex]: [tex]\( r(1) = 3(1) + 2 = 5 \)[/tex] [does not match initial condition]
- This function does not satisfy the criteria.
### 5. [tex]\( r(n)=2n+3 \)[/tex]
- For [tex]\( n = 1 \)[/tex]: [tex]\( r(1) = 2(1) + 3 = 5 \)[/tex] [does not match initial condition]
- This function does not satisfy the criteria.
### 6. [tex]\( r(1)=3 ; r(n)=r(n-1)-1 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- For [tex]\( n = 1 \)[/tex]: [tex]\( r(1) = 3 \)[/tex] [does not match initial condition]
- This function does not satisfy the criteria.
### Conclusion
Based on the above analysis, the functions that correctly determine the number of rings Martha designs are:
[tex]\[ r(1)=2 ; r(n)=r(n-1)+3, \text{ for } n \geq 2 \][/tex]
[tex]\[ r(n)=3n-1 \][/tex]
These functions accurately represent the situation and are consistent with the information given. Therefore, the correct answer is:
[tex]\[ [2] \][/tex]
