High School

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------------------------------------------------ The pattern of numbers below is an arithmetic sequence:

[tex]\[ 14, 24, 34, 44, 54, \ldots \][/tex]

Which statement describes the recursive function used to generate the sequence?

A. The common difference is 1, so the function is [tex]f(n+1)=f(n)+1[/tex] where [tex]f(1)=14[/tex].

B. The common difference is 4, so the function is [tex]f(n+1)=f(n)+4[/tex] where [tex]f(1)=10[/tex].

C. The common difference is 10, so the function is [tex]f(n+1)=f(n)+10[/tex] where [tex]f(1)=14[/tex].

Answer :

To solve this problem, we need to identify the pattern in the given arithmetic sequence and then describe the recursive function that represents it. The sequence given is:

14, 24, 34, 44, 54, ...

Here's a step-by-step approach to understanding the sequence and finding the correct recursive function:

1. Identify the Common Difference:
- An arithmetic sequence is defined by a constant difference between consecutive terms.
- Let's find the difference between the first two terms: 24 - 14 = 10.
- Check the difference between other consecutive terms to ensure consistency:
- 34 - 24 = 10
- 44 - 34 = 10
- 54 - 44 = 10
- The common difference is consistently 10.

2. Define the First Term:
- The first term of the sequence, [tex]\( f(1) \)[/tex], is 14.

3. Describe the Recursive Function:
- A recursive function that generates this sequence takes the form:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- With the common difference identified as 10, the recursive function becomes:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Since the first term is 14, we have:
[tex]\[
f(1) = 14
\][/tex]

4. Select the Correct Statement:
- Based on the common difference of 10 and the first term being 14, the right 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]."

This accurately describes the recursive function used to generate the arithmetic sequence given in the problem.