Answer :
To address the question of determining the correct recursive function for the given arithmetic sequence, let's start by considering the sequence provided:
14, 24, 34, 44, 54, ...
We can treat this as an arithmetic sequence, which means that each term is derived from the previous one by adding a constant value known as the common difference.
### Step 1: Identify the Common Difference
To find the common difference, we subtract the first term from the second term:
[tex]\[ 24 - 14 = 10 \][/tex]
### Step 2: Establish the Recursive Function
Now that we know the common difference is 10, we can form a recursive function. A recursive function describes how each term in a sequence is derived from the previous term. The general form for the recursive function of an arithmetic sequence is:
[tex]\[ f(n+1) = f(n) + \text{common difference} \][/tex]
For this sequence, substituting the common difference, the function becomes:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
### Step 3: Identify the First Term
The first term in the sequence is 14, which means:
[tex]\[ f(1) = 14 \][/tex]
### Conclusion
The correct statement that describes the recursive function used to generate the 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]."
This matches the observations from the sequence and describes how each term is generated consistently.
14, 24, 34, 44, 54, ...
We can treat this as an arithmetic sequence, which means that each term is derived from the previous one by adding a constant value known as the common difference.
### Step 1: Identify the Common Difference
To find the common difference, we subtract the first term from the second term:
[tex]\[ 24 - 14 = 10 \][/tex]
### Step 2: Establish the Recursive Function
Now that we know the common difference is 10, we can form a recursive function. A recursive function describes how each term in a sequence is derived from the previous term. The general form for the recursive function of an arithmetic sequence is:
[tex]\[ f(n+1) = f(n) + \text{common difference} \][/tex]
For this sequence, substituting the common difference, the function becomes:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
### Step 3: Identify the First Term
The first term in the sequence is 14, which means:
[tex]\[ f(1) = 14 \][/tex]
### Conclusion
The correct statement that describes the recursive function used to generate the 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]."
This matches the observations from the sequence and describes how each term is generated consistently.
