Question
Complete the missing values in the recursive formula of the sequence.
\[\begin{cases}
f(1)=A\\\\
f(n)=f(n-1)+B
\end{cases}\]
\[A=\]
\[B=\]
\[\begin{cases}
f(1)=A\\\\
f(n)=f(n-1)+B
\end{cases}\]
\[A=\]
\[B=\]
Answers
GPT-4o mini
To complete the missing values in the recursive formula for the sequence, we can analyze the given information:
1. The recursion specifies that \( f(1) = A \), which means the initial term of the sequence is defined by \( A \).
2. The recursive relation \( f(n) = f(n-1) + B \) shows that each term in the sequence is obtained by adding \( B \) to the previous term \( f(n-1) \).
Given this structure, we can express the first few terms of the sequence:
- \( f(1) = A \)
- \( f(2) = f(1) + B = A + B \)
- \( f(3) = f(2) + B = (A + B) + B = A + 2B \)
- \( f(4) = f(3) + B = (A + 2B) + B = A + 3B \)
This suggests that the general form of the sequence can be formulated as follows:
\[
f(n) = A + (n - 1)B
\]
From this observation, we can conclude:
- \( A \) represents the first term of the sequence (when \( n=1 \)).
- \( B \) represents the common difference between consecutive terms.
Since no specific values are given for \( A \) and \( B \) in your query, we simply state:
\[
A = \text{(initial term of the sequence)}
\]
\[
B = \text{(common difference between terms)}
\]
If you have specific values for \( A \) and \( B \), please provide them so that I can give more targeted information. Otherwise, please choose values based on your requirements for the sequence.
1. The recursion specifies that \( f(1) = A \), which means the initial term of the sequence is defined by \( A \).
2. The recursive relation \( f(n) = f(n-1) + B \) shows that each term in the sequence is obtained by adding \( B \) to the previous term \( f(n-1) \).
Given this structure, we can express the first few terms of the sequence:
- \( f(1) = A \)
- \( f(2) = f(1) + B = A + B \)
- \( f(3) = f(2) + B = (A + B) + B = A + 2B \)
- \( f(4) = f(3) + B = (A + 2B) + B = A + 3B \)
This suggests that the general form of the sequence can be formulated as follows:
\[
f(n) = A + (n - 1)B
\]
From this observation, we can conclude:
- \( A \) represents the first term of the sequence (when \( n=1 \)).
- \( B \) represents the common difference between consecutive terms.
Since no specific values are given for \( A \) and \( B \) in your query, we simply state:
\[
A = \text{(initial term of the sequence)}
\]
\[
B = \text{(common difference between terms)}
\]
If you have specific values for \( A \) and \( B \), please provide them so that I can give more targeted information. Otherwise, please choose values based on your requirements for the sequence.