Answer :
To solve this problem, we need to determine the recursive function that accurately describes the given arithmetic sequence.
The sequence provided is: 14, 24, 34, 44, 54, ...
Step 1: Identify the Common Difference
An arithmetic sequence has a constant difference between consecutive terms, called the "common difference." To find the common difference, subtract the first term from the second term:
[tex]\[ 24 - 14 = 10 \][/tex]
Step 2: Use the Common Difference in the Recursive Function
The general form of a recursive function for an arithmetic sequence is:
[tex]\[ f(n+1) = f(n) + \text{common difference} \][/tex]
Since we found the common difference is 10, we can write:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
Step 3: Determine the Initial Term
The initial term of the sequence is the first term, which is 14. In recursive function language, we represent this as:
[tex]\[ f(1) = 14 \][/tex]
Therefore, the recursive function for this sequence is:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
with the initial condition:
[tex]\[ f(1) = 14 \][/tex]
The correct statement that describes the recursive function used to generate this sequence is:
"The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex]."
The sequence provided is: 14, 24, 34, 44, 54, ...
Step 1: Identify the Common Difference
An arithmetic sequence has a constant difference between consecutive terms, called the "common difference." To find the common difference, subtract the first term from the second term:
[tex]\[ 24 - 14 = 10 \][/tex]
Step 2: Use the Common Difference in the Recursive Function
The general form of a recursive function for an arithmetic sequence is:
[tex]\[ f(n+1) = f(n) + \text{common difference} \][/tex]
Since we found the common difference is 10, we can write:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
Step 3: Determine the Initial Term
The initial term of the sequence is the first term, which is 14. In recursive function language, we represent this as:
[tex]\[ f(1) = 14 \][/tex]
Therefore, the recursive function for this sequence is:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
with the initial condition:
[tex]\[ f(1) = 14 \][/tex]
The correct statement that describes the recursive function used to generate this sequence is:
"The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex]."
