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Answer :
To solve the question about the pattern of numbers in the arithmetic sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], let's break it down step-by-step:
1. Understand the Arithmetic Sequence:
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference."
2. Identify the Common Difference:
To find the common difference, subtract any term from the term that follows it. Let's use the first two terms in the sequence:
[tex]\[
24 - 14 = 10
\][/tex]
So, the common difference is [tex]\(10\)[/tex].
3. Writing the Recursive Formula:
A recursive formula defines each term of the sequence by referring to the previous term. For an arithmetic sequence, the recursive formula is generally expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
For the given sequence, since we determined that the common difference is [tex]\(10\)[/tex] and the first term [tex]\(f(1)\)[/tex] is [tex]\(14\)[/tex], the recursive function can be written as:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Putting it All Together:
Based on the steps and the arithmetic sequence given, 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].
This recursive formula correctly describes how to generate each term of the sequence from the previous one.
1. Understand the Arithmetic Sequence:
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference."
2. Identify the Common Difference:
To find the common difference, subtract any term from the term that follows it. Let's use the first two terms in the sequence:
[tex]\[
24 - 14 = 10
\][/tex]
So, the common difference is [tex]\(10\)[/tex].
3. Writing the Recursive Formula:
A recursive formula defines each term of the sequence by referring to the previous term. For an arithmetic sequence, the recursive formula is generally expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
For the given sequence, since we determined that the common difference is [tex]\(10\)[/tex] and the first term [tex]\(f(1)\)[/tex] is [tex]\(14\)[/tex], the recursive function can be written as:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Putting it All Together:
Based on the steps and the arithmetic sequence given, 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].
This recursive formula correctly describes how to generate each term of the sequence from the previous one.
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