Question
given the recursively defined formula a1 = -3, an = an -1 + 4, determine which option is the correct function notation.
a a(1) = -3; f(n) = f(n-1)+4
b f(1)=-3;f(n)=f(an-1)+4
c f(1)=-3;f(n)=f(n-1)+4
d f(1)=-3;f(n)=an-1+4
a a(1) = -3; f(n) = f(n-1)+4
b f(1)=-3;f(n)=f(an-1)+4
c f(1)=-3;f(n)=f(n-1)+4
d f(1)=-3;f(n)=an-1+4
Answers
Answered by
GPT-4o mini
Let's analyze the recursively defined formula given:
1. We start with \( a_1 = -3 \).
2. The recursive rule is \( a_n = a_{n-1} + 4 \).
Now, let's evaluate each option:
a. \( a(1) = -3; \, f(n) = f(n-1) + 4 \)
- This option uses \( a(1) \), which is not a standard notation for functions. Instead, it should be \( f(1) \). But \( f(n) \) defined as \( f(n-1) + 4 \) follows the same recursion as \( a_n \). This option is incorrect due to the first part.
b. \( f(1) = -3; \, f(n) = f(a_{n-1}) + 4 \)
- This option defines \( f(n) \) in terms of \( a_{n-1} \), which does not match the original definition of \( a_n \) where \( a_n \) is calculated using the previous term \( a_{n-1} \), not a function of itself. Therefore, this option is incorrect.
c. \( f(1) = -3; \, f(n) = f(n-1) + 4 \)
- This option correctly defines the starting point as \( f(1) = -3 \) and follows the same recursion as \( a_n \). This option is consistent with the original definition of \( a_n \) where each term depends on the previous term.
d. \( f(1) = -3; \, f(n) = a_{n-1} + 4 \)
- This option also starts correctly with \( f(1) = -3 \), but uses \( a_{n-1} \) as part of the recursive definition rather than incorporating the function itself. It yields different values than the intended recursion.
Given this analysis, the correct option is:
**c: \( f(1) = -3; \, f(n) = f(n-1) + 4 \)**.
1. We start with \( a_1 = -3 \).
2. The recursive rule is \( a_n = a_{n-1} + 4 \).
Now, let's evaluate each option:
a. \( a(1) = -3; \, f(n) = f(n-1) + 4 \)
- This option uses \( a(1) \), which is not a standard notation for functions. Instead, it should be \( f(1) \). But \( f(n) \) defined as \( f(n-1) + 4 \) follows the same recursion as \( a_n \). This option is incorrect due to the first part.
b. \( f(1) = -3; \, f(n) = f(a_{n-1}) + 4 \)
- This option defines \( f(n) \) in terms of \( a_{n-1} \), which does not match the original definition of \( a_n \) where \( a_n \) is calculated using the previous term \( a_{n-1} \), not a function of itself. Therefore, this option is incorrect.
c. \( f(1) = -3; \, f(n) = f(n-1) + 4 \)
- This option correctly defines the starting point as \( f(1) = -3 \) and follows the same recursion as \( a_n \). This option is consistent with the original definition of \( a_n \) where each term depends on the previous term.
d. \( f(1) = -3; \, f(n) = a_{n-1} + 4 \)
- This option also starts correctly with \( f(1) = -3 \), but uses \( a_{n-1} \) as part of the recursive definition rather than incorporating the function itself. It yields different values than the intended recursion.
Given this analysis, the correct option is:
**c: \( f(1) = -3; \, f(n) = f(n-1) + 4 \)**.