Question

the pattern of numbers below is an arithmetic sequence:

(14, 24, 34, 44, 54, \dots)

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:

Find the common difference

$$ d = 24 - 14 = 10 $$

Identify the initial term

$$ f(1) = 14 $$

Formulate the recursive equation

$$ f(n + 1) = f(n) + 10 \quad \text{where} \quad f(1) = 14 $$

Answer:

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