Answer :
To solve the problem of identifying the recursive function for the given arithmetic sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], we need to follow these steps:
1. Identify the Common Difference:
The sequence is arithmetic, which means the difference between consecutive terms is constant. To find this common difference, subtract any term from its subsequent term.
[tex]\[
24 - 14 = 10
\][/tex]
Therefore, the common difference is 10.
2. Identify the First Term:
The first term of the sequence is given as 14.
3. Formulate the Recursive Function:
A recursive function for an arithmetic sequence can be described as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
Using the values we identified:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
We can then express the recursive function as:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
This matches with the correct statement:
"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 Common Difference:
The sequence is arithmetic, which means the difference between consecutive terms is constant. To find this common difference, subtract any term from its subsequent term.
[tex]\[
24 - 14 = 10
\][/tex]
Therefore, the common difference is 10.
2. Identify the First Term:
The first term of the sequence is given as 14.
3. Formulate the Recursive Function:
A recursive function for an arithmetic sequence can be described as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
Using the values we identified:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
We can then express the recursive function as:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
This matches with the correct statement:
"The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."