Answer :
To solve the problem of identifying the recursive function that describes the arithmetic sequence given by the numbers [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], you can follow these steps:
1. Identify the Sequence:
Look at the sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex].
2. Find the Common Difference:
An arithmetic sequence has a common difference between consecutive terms. You can find it by subtracting the first term from the second term.
[tex]\[
\text{Common Difference} = 24 - 14 = 10
\][/tex]
3. Identify the First Term:
The first term in the sequence is the starting point, which is given as [tex]\(14\)[/tex].
4. Write the Recursive Function:
A recursive function describes how to get from one term to the next in the sequence. For an arithmetic sequence with a common difference [tex]\(d\)[/tex] and a first term [tex]\(f(1)\)[/tex]:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
Given that the common difference is [tex]\(10\)[/tex] and the first term is [tex]\(14\)[/tex], 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 for 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]."
1. Identify the Sequence:
Look at the sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex].
2. Find the Common Difference:
An arithmetic sequence has a common difference between consecutive terms. You can find it by subtracting the first term from the second term.
[tex]\[
\text{Common Difference} = 24 - 14 = 10
\][/tex]
3. Identify the First Term:
The first term in the sequence is the starting point, which is given as [tex]\(14\)[/tex].
4. Write the Recursive Function:
A recursive function describes how to get from one term to the next in the sequence. For an arithmetic sequence with a common difference [tex]\(d\)[/tex] and a first term [tex]\(f(1)\)[/tex]:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
Given that the common difference is [tex]\(10\)[/tex] and the first term is [tex]\(14\)[/tex], 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 for 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]."
