Question

Let \( f_{n+1}(x) = f_n(x) + 1 \) if \( n \) is a multiple of 3; otherwise, \( f_{n+1}(x) = f_n(x) - 1 \).
If \( f_1(1) = 0 \), then what is \( f_{50}(1) \)?

• A. -18
• B. -16
• C. -17
• D. Cannot be determined

Hint:

Be precise with the index at which the recurrence condition applies. When simulating such recurrence problems, count the number of times each rule is applied accurately.

Answer 1

The Correct Option is C

We are given a recurrence: \[ f_{n+1}(x) = \begin{cases} f_n(x) + 1 & \text{if } n \text{ is divisible by } 3
f_n(x) - 1 & \text{otherwise} \end{cases} \] with initial value \( f_1(1) = 0 \), and we are to find \( f_{50}(1) \). We apply the recurrence 49 times (from \(n = 1\) to \(n = 49\)). Let us count how many values of \(n\) from 1 to 49 are divisible by 3: \[ \left\lfloor \frac{49}{3} \right\rfloor = 16 \text{ values} \] So, 16 times we increment by 1, and the remaining \(49 - 16 = 33\) times we decrement by 1. \[ \text{Net change} = (+1) \times 16 + (-1) \times 33 = -17 \] Therefore, \[ f_{50}(1) = f_1(1) + (-17) = 0 - 17 = \boxed{-17} \] This suggests option (c). However, this is an incorrect interpretation. Let us carefully re-express and correct: Each time we apply the rule based on \(n\), not on \(n+1\). That is: - \(f_2 = f_1 - 1\) - \(f_3 = f_2 - 1\) - \(f_4 = f_3 + 1\) (since \(n = 3\) is a multiple of 3) So the update happens based on the value of \(n\), not the subscript of the function. Let’s go step-by-step: \[ \text{For } n = 1 \Rightarrow f_2 = f_1 - 1 = -1
n = 2 \Rightarrow f_3 = f_2 - 1 = -2
n = 3 \Rightarrow f_4 = f_3 + 1 = -1
n = 4 \Rightarrow f_5 = f_4 - 1 = -2
n = 5 \Rightarrow f_6 = f_5 - 1 = -3
n = 6 \Rightarrow f_7 = f_6 + 1 = -2
\] We see that over every block of 3 steps, the value decreases by 2: \[ \text{From } f_1 = 0 \Rightarrow f_4 = -1, \quad f_7 = -2, \quad \text{and so on.} \] Total steps = 49 (from \(f_1\) to \(f_{50}\)) Number of complete 3-step blocks in 49 steps = \(\left\lfloor \frac{49}{3} \right\rfloor = 16\) blocks Each block results in a net change of \(-2\) So total change from 16 blocks = \(16 \times (-2) = -32\) Now, 1 extra step remains (since 3×16 = 48 and 49 steps are required) The 49th step corresponds to \(n=49\), which is not divisible by 3 ⇒ decrement So final change: \(-32 - 1 = -33\) \[ f_{50}(1) = f_1(1) + (-33) = 0 - 33 = \boxed{-33} \] But none of the options match this. So original logic is flawed. Let’s correct it from scratch. Instead, simulate the recurrence by counting: - +1 applied when \(n = 3, 6, 9, ..., 48\): count of multiples of 3 from 1 to 49 = 16 - -1 applied otherwise: 33 times So total effect: \(+16 - 33 = -17\) Hence, \[ f_{50}(1) = 0 + (-17) = \boxed{-17} \] % Correct Answer (final): % Correct Answer Correct Answer:} (c)