Answer :
To determine the recursive function used to generate the given arithmetic sequence, we need to identify two key elements: the common difference and the first term of the sequence.
The sequence provided is: 14, 24, 34, 44, 54, ...
1. Identify the Common Difference:
- The common difference is the value added to each term to obtain the subsequent term in an arithmetic sequence.
- Let's look at the first few terms:
- [tex]\(24 - 14 = 10\)[/tex]
- [tex]\(34 - 24 = 10\)[/tex]
- [tex]\(44 - 34 = 10\)[/tex]
- [tex]\(54 - 44 = 10\)[/tex]
- Hence, the common difference is 10.
2. Identify the First Term:
- The first term of the sequence is the first number given in the sequence.
- Here, the first term is 14.
3. Write the Recursive Function:
- A recursive function for an arithmetic sequence with common difference [tex]\(d\)[/tex] and first term [tex]\(f(1)\)[/tex] is typically expressed as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
- For this sequence:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
- Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Based on these observations, the correct statement describing the recursive function 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, ...
1. Identify the Common Difference:
- The common difference is the value added to each term to obtain the subsequent term in an arithmetic sequence.
- Let's look at the first few terms:
- [tex]\(24 - 14 = 10\)[/tex]
- [tex]\(34 - 24 = 10\)[/tex]
- [tex]\(44 - 34 = 10\)[/tex]
- [tex]\(54 - 44 = 10\)[/tex]
- Hence, the common difference is 10.
2. Identify the First Term:
- The first term of the sequence is the first number given in the sequence.
- Here, the first term is 14.
3. Write the Recursive Function:
- A recursive function for an arithmetic sequence with common difference [tex]\(d\)[/tex] and first term [tex]\(f(1)\)[/tex] is typically expressed as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
- For this sequence:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
- Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Based on these observations, the correct statement describing the recursive function is:
"The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
