The function f is defined on the positive integers as follows: f(1) = 5, and for n ≥ 1, f(n+1) = 3f(n) + 1 if f(n) is odd f(n+1) = 12f(n) if f(n) is even The function g is defined on the positive integers as follows: g(1) = 3, and for n ≥ 1, g(n+1) = g(n) + 5 if g(n) is odd g(n+1) = 12g(n) if g(n) is even What is the value of f(1000) − g(1000)?
- A−6
- B−5
- C1
- D2
- E4
- F8
Show the answer and worked solution
answer · D
- A−6
- B−5
- C1
- D2
- E4
- F8
Generate terms until each sequence repeats. For f: 5, 16, 8, 4, 2, 1, 4, 2, 1, …, so from n = 4 onwards it cycles through 4, 2, 1 with period 3, taking the value 4 whenever n leaves remainder 1 on division by 3. As 1000 = 3×333 + 1, f(1000) = 4. For g: 3, 8, 4, 2, 1, 6, 3, …, a cycle of length 6 starting at n = 1; as 1000 = 6×166 + 4, g(1000) = g(4) = 2. The difference is 4 − 2 = 2.