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.

Answer 1

To describe the recursive function used to generate the given arithmetic sequence (14, 24, 34, 44, 54, ...), we first need to determine the common difference of the sequence.

The difference between each consecutive pair of numbers is:
- 24 - 14 = 10
- 34 - 24 = 10
- 44 - 34 = 10
- 54 - 44 = 10

So, the common difference is 10.

To express this in a recursive function form, we can denote the function as \( f(n) \), where:
- \( f(1) = 14 \) (the first term)
- \( f(n + 1) = f(n) + 10 \) (adding the common difference of 10 to the previous term)

Now let's evaluate the statements given:

1. The common difference is 1, so the function is \( f(n + 1) = f(n) + 1 \) where \( f(1) = 14 \). (Incorrect: The common difference is 10, not 1.)
2. The common difference is 4, so the function is \( f(n + 1) = f(n) + 4 \) where \( f(1) = 10 \). (Incorrect: The common difference is 10, not 4, and the first term is 14, not 10.)
3. The common difference is 10, so the function is \( f(n + 1) = f(n) + 10 \) where \( f(1) = 14 \). (Correct)
4. The common difference is 14, so the function is \( f(n + 1) = f(n) + 14 \) where \( f(1) = 10 \). (Incorrect: The common difference is 10, not 14, and the first term is 14, not 10.)

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