Answer :
To determine the correct functions that describe the number of rings [tex]\( r(n) \)[/tex] that Martha designs in [tex]\( n \)[/tex] hours, let's carefully analyze each option provided.
### Analysis of Each Function
1. [tex]\( r(n) = 3n - 1 \)[/tex]
- This function suggests that the number of rings is directly proportional to the number of hours, scaled by a factor of 3, and then reduced by 1.
- For [tex]\( n = 1 \)[/tex], [tex]\( r(1) = 3 \times 1 - 1 = 2 \)[/tex]. This matches with the initial condition.
- For [tex]\( n = 2 \)[/tex], [tex]\( r(2) = 3 \times 2 - 1 = 5 \)[/tex]. Since Martha designs 2 rings in the first hour and 3 additional rings each hour thereafter, the total for 2 hours should be [tex]\( 2 + 3 = 5 \)[/tex]. This also matches.
- This function appears to be consistent with the description.
2. [tex]\( r(1) = 2; r(n) = r(n-1) + 3 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- This is a piecewise recursive function. It specifies that in the first hour, Martha designs 2 rings, and for each subsequent hour, she adds 3 more rings to the previous total.
- For [tex]\( n = 1 \)[/tex], [tex]\( r(1) = 2 \)[/tex]. This matches the initial condition.
- For [tex]\( n = 2 \)[/tex], [tex]\( r(2) = r(1) + 3 = 2 + 3 = 5 \)[/tex]. This matches what we expect (2 rings in the first hour, 3 more in the next).
- This recursive function is consistent with the problem's description.
3. [tex]\( r(n) = 3 \pi + 2 \)[/tex]
- This function involves [tex]\(\pi\)[/tex], which is irrelevant in this context as it introduces an irrational factor unrelated to the problem.
- [tex]\( \pi \)[/tex] does not fit within the hours context described in the problem where [tex]\( n \)[/tex] is an integer.
- This function is not appropriate for the given situation.
4. [tex]\( r(\pi) = 2\pi + 3 \)[/tex]
- Similar to the previous function, this involves [tex]\(\pi\)[/tex], and again, it does not make sense in the context of hours.
- This function is not applicable.
5. [tex]\( \tau(1) = 3; r(n) = r(n-1) + 2 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- The notation starts with [tex]\(\tau\)[/tex] instead of [tex]\( r \)[/tex], which is inconsistent.
- The initial condition, [tex]\( r(1) = 3 \)[/tex], does not match the given condition where [tex]\( r(1) = 2 \)[/tex].
- This function is not consistent with the given problem.
6. [tex]\( r(1) = 3; r(n) = r(n-1) - 1 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- This function starts with an initial condition of [tex]\( r(1) = 3 \)[/tex], which is also inconsistent with [tex]\( r(1) = 2 \)[/tex].
- Reducing rings per hour [tex]\( r(n) = r(n-1) - 1 \)[/tex] doesn't fit the description where she designs 3 additional rings each hour.
- This function is incorrect.
### Conclusion
Based on our analysis, the functions that correctly describe the number of rings Martha designs in [tex]\( n \)[/tex] hours are:
- [tex]\( r(1) = 2; r(n) = r(n-1) + 3 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
These options meet all the given conditions accurately.
### Analysis of Each Function
1. [tex]\( r(n) = 3n - 1 \)[/tex]
- This function suggests that the number of rings is directly proportional to the number of hours, scaled by a factor of 3, and then reduced by 1.
- For [tex]\( n = 1 \)[/tex], [tex]\( r(1) = 3 \times 1 - 1 = 2 \)[/tex]. This matches with the initial condition.
- For [tex]\( n = 2 \)[/tex], [tex]\( r(2) = 3 \times 2 - 1 = 5 \)[/tex]. Since Martha designs 2 rings in the first hour and 3 additional rings each hour thereafter, the total for 2 hours should be [tex]\( 2 + 3 = 5 \)[/tex]. This also matches.
- This function appears to be consistent with the description.
2. [tex]\( r(1) = 2; r(n) = r(n-1) + 3 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- This is a piecewise recursive function. It specifies that in the first hour, Martha designs 2 rings, and for each subsequent hour, she adds 3 more rings to the previous total.
- For [tex]\( n = 1 \)[/tex], [tex]\( r(1) = 2 \)[/tex]. This matches the initial condition.
- For [tex]\( n = 2 \)[/tex], [tex]\( r(2) = r(1) + 3 = 2 + 3 = 5 \)[/tex]. This matches what we expect (2 rings in the first hour, 3 more in the next).
- This recursive function is consistent with the problem's description.
3. [tex]\( r(n) = 3 \pi + 2 \)[/tex]
- This function involves [tex]\(\pi\)[/tex], which is irrelevant in this context as it introduces an irrational factor unrelated to the problem.
- [tex]\( \pi \)[/tex] does not fit within the hours context described in the problem where [tex]\( n \)[/tex] is an integer.
- This function is not appropriate for the given situation.
4. [tex]\( r(\pi) = 2\pi + 3 \)[/tex]
- Similar to the previous function, this involves [tex]\(\pi\)[/tex], and again, it does not make sense in the context of hours.
- This function is not applicable.
5. [tex]\( \tau(1) = 3; r(n) = r(n-1) + 2 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- The notation starts with [tex]\(\tau\)[/tex] instead of [tex]\( r \)[/tex], which is inconsistent.
- The initial condition, [tex]\( r(1) = 3 \)[/tex], does not match the given condition where [tex]\( r(1) = 2 \)[/tex].
- This function is not consistent with the given problem.
6. [tex]\( r(1) = 3; r(n) = r(n-1) - 1 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- This function starts with an initial condition of [tex]\( r(1) = 3 \)[/tex], which is also inconsistent with [tex]\( r(1) = 2 \)[/tex].
- Reducing rings per hour [tex]\( r(n) = r(n-1) - 1 \)[/tex] doesn't fit the description where she designs 3 additional rings each hour.
- This function is incorrect.
### Conclusion
Based on our analysis, the functions that correctly describe the number of rings Martha designs in [tex]\( n \)[/tex] hours are:
- [tex]\( r(1) = 2; r(n) = r(n-1) + 3 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
These options meet all the given conditions accurately.
