Question

the pattern of numbers below is an arithmetic sequence:
14, 24, 34, 44, 54, ...
which statement describes the recursive function used to generate the sequence?
○ the common difference is 1, so the function is $f(n + 1) = f(n) + 1$ where $f(1) = 14$.
○ the common difference is 4, so the function is $f(n + 1) = f(n) + 4$ where $f(1) = 10$.
○ the common difference is 10, so the function is $f(n + 1) = f(n) + 10$ where $f(1) = 14$.
○ the common difference is 14, so the function is $f(n + 1) = f(n) + 14$ where $f(1) = 10$.

Explanation:

1. First, find the common difference of the arithmetic sequence. For the sequence \(14, 24, 34, 44, 54, \dots\), calculate the difference between consecutive terms: \(24 - 14 = 10\), \(34 - 24 = 10\), \(44 - 34 = 10\), \(54 - 44 = 10\). So the common difference \(d = 10\).
2. The first term \(f(1)\) of the sequence is \(14\) (since the first number in the sequence is \(14\)).
3. Now, analyze each option:

• Option 1: The common difference is calculated as \(1\), which is incorrect (we found it's \(10\)). Also, the function \(f(n + 1)=f(n)+1\) would not generate the sequence. Eliminate this option.
• Option 2: The common difference is stated as \(4\), which is incorrect (it's \(10\)). Also, the first term \(f(1) = 10\) is incorrect (the first term is \(14\)). Eliminate this option.
• Option 3: The common difference is \(10\), the function is \(f(n + 1)=f(n)+10\), and \(f(1)=14\). This matches our calculations.
• Option 4: The common difference is stated as \(14\), which is incorrect (it's \(10\)). Also, the first term \(f(1) = 10\) is incorrect. Eliminate this option.

Answer:

The common difference is 10, so the function is \( f(n + 1) = f(n) + 10 \) where \( f(1) = 14 \).