Problem 20 · 1996 AJHSME
Hard
Algebra & Patterns
find-the-cycle
A special key on a calculator replaces the displayed number x with 1 ÷ (1 − x). (For example, from 2 it gives 1 ÷ (1 − 2) = −1.) If the calculator shows 5 and the key is pressed 100 times in a row, the calculator will display
Show answer
Answer: A — −0.25.
Show hints
Hint 1 of 2
100 presses is way too many to do by hand — but you don't have to. Press the key a few times and watch what happens to the numbers. These repeat-an-operation problems almost always loop back on themselves.
Still stuck? Show hint 2 →
Hint 2 of 2
Once a value you've already seen comes back, you've found a CYCLE. Count its length, then figure out where press #100 lands by seeing how many full cycles fit and what's left over.
Show solution
Approach: find the cycle, then reduce 100 by its length
- Press from 5 and track: 5 → 1 ÷ (1 − 5) = −0.25 → 1 ÷ (1 + 0.25) = 0.8 → 1 ÷ (1 − 0.8) = 5. It's back to 5 after 3 presses — a cycle of length 3.
- So presses 3, 6, 9, … (every multiple of 3) return to 5. Since 100 = 3·33 + 1, press #100 is 1 step past a full cycle — the same as press #1: −0.25.
- Why this transfers: for any 'apply this rule N times' problem with N large, hunt for a repeating cycle, then take N's leftover after dividing by the cycle length. The huge count collapses to a tiny one.
Mark:
· log in to save