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].

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

Answer :

To solve this problem, we first need to understand what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the "common difference."

Let's analyze the given sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex]

### Step 1: Find the Common Difference

To find the common difference, subtract the first term from the second term:

[tex]\[
24 - 14 = 10
\][/tex]

Now, check if the difference between consecutive terms remains consistent:

[tex]\[
34 - 24 = 10, \quad 44 - 34 = 10, \quad 54 - 44 = 10
\][/tex]

The common difference is 10.

### Step 2: Define the Recursive Function

A recursive function for an arithmetic sequence is of the form:

[tex]\[
f(n+1) = f(n) + d
\][/tex]

where [tex]\(d\)[/tex] is the common difference, and [tex]\(f(1)\)[/tex] is the first term of the sequence. For this sequence:

- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.

Therefore, the recursive function is:

[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]

### Step 3: Compare to Given Options

Now, let's compare this to the options provided:

1. The common difference is 1, so the function is [tex]\(f(n+1) = f(n) + 1\)[/tex] where [tex]\(f(1) = 14\)[/tex].
This is incorrect because the common difference is not 1.

2. The common difference is 4, so the function is [tex]\(f(n+1) = f(n) + 4\)[/tex] where [tex]\(f(1) = 10\)[/tex].
This is incorrect because the common difference is not 4, and the first term is not 10.

3. The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex].
This is correct because both the common difference and the first term match our calculation.

4. The common difference is 14, so the function is [tex]\(f(n+1) = f(n) + 14\)[/tex] where [tex]\(f(1) = 10\)[/tex].
This is incorrect because the common difference is not 14, and the first term is not 10.

The correct answer 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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