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Answer :
Certainly! Let's solve the problem step by step to identify the correct recursive function statement for the given arithmetic sequence: 14, 24, 34, 44, 54, ...
1. Identify the Sequence Type:
This is an arithmetic sequence, which means each term is obtained by adding a fixed number, called the common difference, to the previous term.
2. Find the Common Difference:
To find the common difference, subtract the first term from the second term in the sequence:
[tex]\( 24 - 14 = 10 \)[/tex]
Therefore, the common difference is 10.
3. Understand the Recursive Function:
A recursive function for an arithmetic sequence with the common difference [tex]\(d\)[/tex] is generally expressed as:
[tex]\( f(n+1) = f(n) + d \)[/tex]
In this case, given that the common difference [tex]\(d = 10\)[/tex], the function is:
[tex]\( f(n+1) = f(n) + 10 \)[/tex]
4. Identify the First Term:
The first term of the sequence is given as 14, so:
[tex]\( f(1) = 14 \)[/tex]
5. Choose the Correct Statement:
Now, we can choose the statement that matches our findings:
- The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
Thus, the correct statement describing the recursive function used to generate the arithmetic 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]."
1. Identify the Sequence Type:
This is an arithmetic sequence, which means each term is obtained by adding a fixed number, called the common difference, to the previous term.
2. Find the Common Difference:
To find the common difference, subtract the first term from the second term in the sequence:
[tex]\( 24 - 14 = 10 \)[/tex]
Therefore, the common difference is 10.
3. Understand the Recursive Function:
A recursive function for an arithmetic sequence with the common difference [tex]\(d\)[/tex] is generally expressed as:
[tex]\( f(n+1) = f(n) + d \)[/tex]
In this case, given that the common difference [tex]\(d = 10\)[/tex], the function is:
[tex]\( f(n+1) = f(n) + 10 \)[/tex]
4. Identify the First Term:
The first term of the sequence is given as 14, so:
[tex]\( f(1) = 14 \)[/tex]
5. Choose the Correct Statement:
Now, we can choose the statement that matches our findings:
- The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
Thus, the correct statement describing the recursive function used to generate the arithmetic 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]."
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