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Answer :
To determine the recursive function that generates the given arithmetic sequence, let's break it down step-by-step.
1. Identify the Sequence Provided:
The sequence in question is: 14, 24, 34, 44, 54, ...
2. Calculate the Common Difference:
In an arithmetic sequence, each term after the first is found by adding a constant called the common difference to the previous term. We can find this common difference by subtracting any term from the next term in the sequence.
Let's calculate using the first two numbers in the sequence:
- [tex]\(24 - 14 = 10\)[/tex]
- Likewise, [tex]\(34 - 24 = 10\)[/tex]
- This confirms the pattern that the common difference is 10.
3. Determine the Recurrence Relation:
Now that we know the common difference, we can write the recursive function. A recursive function defines each term in the sequence based on its previous term using this formula:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
Substituting the value we found:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
4. Specify the Initial Term:
For a recursive function, we also need to know the first term of the sequence, which is given as 14. Therefore, the initial condition is:
[tex]\[
f(1) = 14
\][/tex]
5. Putting It All Together:
Therefore, the recursive function to generate this sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
This function correctly describes the rules for generating the sequence provided.
1. Identify the Sequence Provided:
The sequence in question is: 14, 24, 34, 44, 54, ...
2. Calculate the Common Difference:
In an arithmetic sequence, each term after the first is found by adding a constant called the common difference to the previous term. We can find this common difference by subtracting any term from the next term in the sequence.
Let's calculate using the first two numbers in the sequence:
- [tex]\(24 - 14 = 10\)[/tex]
- Likewise, [tex]\(34 - 24 = 10\)[/tex]
- This confirms the pattern that the common difference is 10.
3. Determine the Recurrence Relation:
Now that we know the common difference, we can write the recursive function. A recursive function defines each term in the sequence based on its previous term using this formula:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
Substituting the value we found:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
4. Specify the Initial Term:
For a recursive function, we also need to know the first term of the sequence, which is given as 14. Therefore, the initial condition is:
[tex]\[
f(1) = 14
\][/tex]
5. Putting It All Together:
Therefore, the recursive function to generate this sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
This function correctly describes the rules for generating the sequence provided.
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