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Escape Room · Geometric Sequences
The Lab Report
State Science Competition · Submission Portal · Closes midnight
← Notes
Samples verified: 0 of 6 0%
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📍 Scene Setup The State Science Competition submission portal locks at midnight. Josh's biology partner has recorded a bacterial growth experiment, but the data file got corrupted, only the first five hourly readings survived.

The competition entry requires six verified calculations from the dataset before the portal will accept the report. Josh needs to reconstruct the missing values, run the analysis, and submit before the deadline.

🧫 Bacterial Colony, Hourly Count (cells × 10³)

Hour 1
5
Hour 2
20
Hour 3
80
Hour 4
320
Hour 5
1280
Hour 6
?
First term a = 5  ·  Geometric sequence  ·  Formula: tₙ = a × rn−1
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Sample 1, Common Ratio
The bacterial counts are: 5, 20, 80, 320, 1280, ...

What is the common ratio (r) for this geometric sequence?
Divide any term by the one before it, the answer should be the same every time.
💡 r = t₂ ÷ t₁ = 20 ÷ 5 = ?
✅ r = 4. The colony quadruples every hour, rapid exponential growth.
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Sample 2, Predict Hour 6
Using a = 5 and r = 4, what is the bacterial count at hour 6?
Use: tₙ = a × rⁿ⁻¹. Substitute n = 6.
💡 t₆ = 5 × 4⁵ = 5 × 1024 = ?
✅ Hour 6 = 5120. The colony is growing fast, exponential growth in action!
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Sample 3, Total Count
What is the total bacterial count across the first 4 hours of the experiment? (Find S₄.)
Use: Sₙ = a(rⁿ − 1) ÷ (r − 1). Here n = 4, a = 5, r = 4.
💡 S₄ = 5 × (4⁴ − 1) ÷ (4 − 1) = 5 × (256 − 1) ÷ 3 = 5 × 255 ÷ 3 = ?
✅ S₄ = 425 thousand cells. The total accumulates quickly with geometric growth.
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Sample 4, Identify the Term
At which hour does the count first reach 1280?
Set tₙ = 1280 and solve: 5 × 4ⁿ⁻¹ = 1280. Divide by 5, then ask: 4 to what power equals that?
💡 5 × 4ⁿ⁻¹ = 1280 → 4ⁿ⁻¹ = 256 → 4ⁿ⁻¹ = 4⁴ → n−1 = 4 → n = ?
✅ Hour 5! That matches the data, good cross-check for the report.
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Sample 5, Second Culture
A second bacterial culture starts with a = 2 and has a common ratio of r = 3.

What is the count at hour 4?
Use the same formula: t₄ = a × r³
💡 t₄ = 2 × 3⁴⁻¹ = 2 × 3³ = 2 × 27 = ?
✅ t₄ = 54. Culture B grows more slowly, smaller r means slower spread.
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Sample 6, Work Backwards
A third culture has t₁ = 7 and t₄ = 189.

What is the common ratio?
Use t₄ = a × r³. Substitute and solve for r.
💡 7 × r³ = 189 → r³ = 189 ÷ 7 = 27 → r = ∛27 = ?
✅ r = 3. Culture C: starting at 7, tripling each hour, the final piece of the report.
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Submission Accepted!
All 6 samples verified. The competition portal has accepted Josh's lab report with 4 seconds to spare. The bacterial growth data checks out perfectly, every calculation lines up with the geometric model.

r = 4, t₆ = 5120, S₄ = 425, hour 5 = 1280, culture B t₄ = 54, culture C r = 3. 🏆
tₙ = a × rⁿ⁻¹
Sₙ = a(rⁿ−1) ÷ (r−1)
r = t₂ ÷ t₁
r = ⁿ⁻¹√(tₙ ÷ a)
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