Answer :
To solve the problem of finding the recursive function for the given arithmetic sequence, let's break it down step by step.
The sequence given is: 14, 24, 34, 44, 54, ...
Step 1: Identify the First Term
The first term of this sequence is easy to identify. It is the number at the start of the sequence:
- First term, [tex]\( f(1) = 14 \)[/tex].
Step 2: Determine the Common Difference
The common difference is the amount added to each term to get to the next term. We find this by subtracting any term from the term that follows it:
- For example: [tex]\( 24 - 14 = 10 \)[/tex].
Repeating this for the next terms: [tex]\( 34 - 24 = 10 \)[/tex], [tex]\( 44 - 34 = 10 \)[/tex], and so on, confirms the common difference.
- Common difference, [tex]\( d = 10 \)[/tex].
Step 3: Write the Recursive Function
Using the first term and the common difference, we can write the recursive function for the sequence. The general form of a recursive function for an arithmetic sequence is:
[tex]\[ f(n+1) = f(n) + d \][/tex]
Applying it to our sequence:
- Since [tex]\( f(1) = 14 \)[/tex] and the common difference [tex]\( d = 10 \)[/tex], the recursive function becomes:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
- Where [tex]\( f(1) = 14 \)[/tex].
This describes how each subsequent term in the sequence is generated by adding 10 to the previous term.
Conclusion
The 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].
The sequence given is: 14, 24, 34, 44, 54, ...
Step 1: Identify the First Term
The first term of this sequence is easy to identify. It is the number at the start of the sequence:
- First term, [tex]\( f(1) = 14 \)[/tex].
Step 2: Determine the Common Difference
The common difference is the amount added to each term to get to the next term. We find this by subtracting any term from the term that follows it:
- For example: [tex]\( 24 - 14 = 10 \)[/tex].
Repeating this for the next terms: [tex]\( 34 - 24 = 10 \)[/tex], [tex]\( 44 - 34 = 10 \)[/tex], and so on, confirms the common difference.
- Common difference, [tex]\( d = 10 \)[/tex].
Step 3: Write the Recursive Function
Using the first term and the common difference, we can write the recursive function for the sequence. The general form of a recursive function for an arithmetic sequence is:
[tex]\[ f(n+1) = f(n) + d \][/tex]
Applying it to our sequence:
- Since [tex]\( f(1) = 14 \)[/tex] and the common difference [tex]\( d = 10 \)[/tex], the recursive function becomes:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
- Where [tex]\( f(1) = 14 \)[/tex].
This describes how each subsequent term in the sequence is generated by adding 10 to the previous term.
Conclusion
The 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].
