🎮
Escape Room · Recurrence Relations
The Score Tracker
Strategy Tournament · Scoring Engine · Leaderboard locks at midnight
← Notes
Rounds verified: 0 of 6 0%
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📍 Scene Setup Josh is ranked #3 in the state for an online strategy game. The tournament scoring engine uses recurrence relations to calculate how resources grow each round. Tonight's leaderboard finalises at midnight, but the scoring system has flagged Josh's submitted calculations as unverified.

To confirm his score, he needs to manually step through 6 rounds of calculations. Get them right and the leaderboard accepts his entry.

🎯 Main Game, Resource Tracker

Round 1
200
Round 2
?
Round 3
?
Round 4
?
Round 5+
...
Game rule: Each round, resources double, then 100 are spent on upgrades.
Recurrence: tₙ₊₁ = 2 × tₙ − 100  ·  Starting value: t₁ = 200
1
Round 1 → 2
Using the recurrence tₙ₊₁ = 2 × tₙ − 100 and t₁ = 200, what is the resource count at the start of round 2?
Substitute t₁ into the rule and calculate.
💡 t₂ = 2 × t₁ − 100 = 2 × 200 − 100 = 400 − 100 = ?
✅ t₂ = 300. Resources doubled then 100 spent on upgrades.
2
Round 2 → 3
Now use t₂ = 300 to find the resources at the start of round 3.
Same rule: tₙ₊₁ = 2 × tₙ − 100. Just feed in the new current value.
💡 t₃ = 2 × 300 − 100 = 600 − 100 = ?
✅ t₃ = 500. The sequence: 200 → 300 → 500 → building fast.
3
Round 3 → 4
Continue the chain. What are the resources at the start of round 4?
tₙ₊₁ = 2 × tₙ − 100. Use the value you just found.
💡 t₄ = 2 × 500 − 100 = 1000 − 100 = ?
✅ t₄ = 900. The main game sequence is 200, 300, 500, 900, the jumps keep getting bigger.
4
Bonus Track, Arithmetic Recurrence
A different player's resource tracker uses: tₙ₊₁ = tₙ + 60, with t₁ = 40.

What are their resources at round 4?
This is an arithmetic recurrence (r = 1). Just add 60 each time: t₁ → t₂ → t₃ → t₄.
💡 t₂ = 100, t₃ = 160, t₄ = ?   (add 60 each time)
✅ t₄ = 220. Arithmetic recurrence, steady linear growth.
5
Bonus Track, Geometric Recurrence
Another tracker: tₙ₊₁ = 3 × tₙ, t₁ = 10.

What is the round 4 resource count?
Geometric recurrence (d = 0). Multiply by 3 each round: t₁ → t₂ → t₃ → t₄.
💡 t₂ = 30, t₃ = 90, t₄ = ?   (multiply by 3 each time)
✅ t₄ = 270. Geometric recurrence, exponential growth, no spending.
6
Final Round, Financial Recurrence
The tournament prize pool starts at $500. Each week it grows by 20%, and the organisers add a $100 sponsorship contribution.

Using tₙ₊₁ = 1.2 × tₙ + 100 and t₁ = 500, what is t₃ (the prize pool after 2 growth cycles)?
Generate t₂ first, then use it to find t₃. Two steps.
💡 t₂ = 1.2 × 500 + 100 = 700. Now: t₃ = 1.2 × 700 + 100 = ?
✅ t₃ = $940. Both r and d at work, this is what financial sequences look like.
🎮
Leaderboard Updated!
All 6 rounds verified. The scoring engine has accepted Josh's calculations and updated the leaderboard with 12 seconds to spare. Rank confirmed: #3 in the state.

Arithmetic, geometric, financial, three types of recurrence, all handled. The tournament commences tomorrow. 🏆
tₙ₊₁ = r·tₙ + d
r=1, d≠0 → arithmetic
r≠1, d=0 → geometric
r≠1, d≠0 → financial
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