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].
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].
