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Answer :
To solve this problem, let's look at the arithmetic sequence provided: 14, 24, 34, 44, 54, ...
In an arithmetic sequence, each term after the first is obtained by adding a constant difference to the previous term. This constant difference is called the "common difference."
Step-by-step solution:
1. Identify the common difference:
To find the common difference, subtract the first term from the second term.
[tex]\[
\text{Second term} - \text{First term} = 24 - 14 = 10
\][/tex]
Thus, the common difference is 10.
2. Write the recursive function:
In a recursive sequence, each term is defined in terms of the previous term. A general recursive formula for an arithmetic sequence is:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference. In this sequence:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
3. Determine the initial term:
The first term of the sequence (when [tex]\(n=1\)[/tex]) is given as 14:
[tex]\[
f(1) = 14
\][/tex]
4. Select the correct statement:
With the common difference identified as 10 and the first term as 14, the appropriate recursive function that represents the arithmetic sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Given this information, the statement that matches 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 third option correctly describes the recursive function used to generate the sequence.
In an arithmetic sequence, each term after the first is obtained by adding a constant difference to the previous term. This constant difference is called the "common difference."
Step-by-step solution:
1. Identify the common difference:
To find the common difference, subtract the first term from the second term.
[tex]\[
\text{Second term} - \text{First term} = 24 - 14 = 10
\][/tex]
Thus, the common difference is 10.
2. Write the recursive function:
In a recursive sequence, each term is defined in terms of the previous term. A general recursive formula for an arithmetic sequence is:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference. In this sequence:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
3. Determine the initial term:
The first term of the sequence (when [tex]\(n=1\)[/tex]) is given as 14:
[tex]\[
f(1) = 14
\][/tex]
4. Select the correct statement:
With the common difference identified as 10 and the first term as 14, the appropriate recursive function that represents the arithmetic sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Given this information, the statement that matches 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 third option correctly describes the recursive function used to generate the sequence.
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