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Answer :
To determine the recursive function that generates the given sequence, we first need to identify the pattern in the sequence of numbers: 14, 24, 34, 44, 54, ...
1. Identify the Common Difference:
- The sequence is an arithmetic sequence, which means each term is obtained by adding a constant value, known as the common difference, to the preceding term.
- Let's find the common difference by subtracting the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
- So, the common difference is 10.
2. Determine the First Term:
- The first term in the sequence is 14, so [tex]\( f(1) = 14 \)[/tex].
3. Write the Recursive Function:
- In a recursive formula for an arithmetic sequence, the next term [tex]\( f(n+1) \)[/tex] is given by the sum of the current term [tex]\( f(n) \)[/tex] and the common difference:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Knowing the first term, we can state that initially, [tex]\( f(1) = 14 \)[/tex].
Given these observations, the statement that describes the recursive function used to generate 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 Common Difference:
- The sequence is an arithmetic sequence, which means each term is obtained by adding a constant value, known as the common difference, to the preceding term.
- Let's find the common difference by subtracting the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
- So, the common difference is 10.
2. Determine the First Term:
- The first term in the sequence is 14, so [tex]\( f(1) = 14 \)[/tex].
3. Write the Recursive Function:
- In a recursive formula for an arithmetic sequence, the next term [tex]\( f(n+1) \)[/tex] is given by the sum of the current term [tex]\( f(n) \)[/tex] and the common difference:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Knowing the first term, we can state that initially, [tex]\( f(1) = 14 \)[/tex].
Given these observations, the statement that describes the recursive function used to generate 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]."
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