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Answer :
To determine the recursive function used to generate the given arithmetic sequence [tex]\(14,24,34,44,54, \ldots\)[/tex], we need to identify the common difference and the initial term of the sequence. Here is the step-by-step solution:
1. Identify the initial term (first term):
The first term of the given sequence is [tex]\( f(1) = 14 \)[/tex].
2. Determine the common difference:
The common difference in an arithmetic sequence is the difference between any two consecutive terms.
- [tex]\( 24 - 14 = 10 \)[/tex]
- [tex]\( 34 - 24 = 10 \)[/tex]
- [tex]\( 44 - 34 = 10 \)[/tex]
- [tex]\( 54 - 44 = 10 \)[/tex]
Therefore, the common difference [tex]\(d\)[/tex] is [tex]\(10\)[/tex].
3. Formulate the recursive function:
In an arithmetic sequence, the recursive function can be written as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\( f(n) \)[/tex] is the nth term, and [tex]\( d \)[/tex] is the common difference.
Substituting the initial term and the common difference into the formula, we get:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
with the initial term:
[tex]\[
f(1) = 14
\][/tex]
4. Select the correct statement:
From the given statements, the one that matches our formulation is:
- "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 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 initial term (first term):
The first term of the given sequence is [tex]\( f(1) = 14 \)[/tex].
2. Determine the common difference:
The common difference in an arithmetic sequence is the difference between any two consecutive terms.
- [tex]\( 24 - 14 = 10 \)[/tex]
- [tex]\( 34 - 24 = 10 \)[/tex]
- [tex]\( 44 - 34 = 10 \)[/tex]
- [tex]\( 54 - 44 = 10 \)[/tex]
Therefore, the common difference [tex]\(d\)[/tex] is [tex]\(10\)[/tex].
3. Formulate the recursive function:
In an arithmetic sequence, the recursive function can be written as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\( f(n) \)[/tex] is the nth term, and [tex]\( d \)[/tex] is the common difference.
Substituting the initial term and the common difference into the formula, we get:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
with the initial term:
[tex]\[
f(1) = 14
\][/tex]
4. Select the correct statement:
From the given statements, the one that matches our formulation is:
- "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 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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