Answer :
To solve this problem, we need to determine the recursive function that describes the arithmetic sequence provided: 14, 24, 34, 44, 54, ...
An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant, called the common difference.
Step-by-step Solution:
1. Identify the Sequence:
The sequence given is: 14, 24, 34, 44, 54, ...
2. Calculate the Common Difference:
To find the common difference, subtract each term from the term that follows it:
- 24 - 14 = 10
- 34 - 24 = 10
- 44 - 34 = 10
- 54 - 44 = 10
The common difference is consistent and equals 10.
3. Formulate the Recursive Function:
A recursive function for an arithmetic sequence has the form:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference and [tex]\(f(1)\)[/tex] is the first term of the sequence.
4. Apply the Values:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
Therefore, the recursive function for this sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Based on the information and calculations, we conclude that the correct statement is: "The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant, called the common difference.
Step-by-step Solution:
1. Identify the Sequence:
The sequence given is: 14, 24, 34, 44, 54, ...
2. Calculate the Common Difference:
To find the common difference, subtract each term from the term that follows it:
- 24 - 14 = 10
- 34 - 24 = 10
- 44 - 34 = 10
- 54 - 44 = 10
The common difference is consistent and equals 10.
3. Formulate the Recursive Function:
A recursive function for an arithmetic sequence has the form:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference and [tex]\(f(1)\)[/tex] is the first term of the sequence.
4. Apply the Values:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
Therefore, the recursive function for this sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Based on the information and calculations, we conclude that the correct statement is: "The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
