📍 Scene Setup
Josh has been training for the Coastal Fun Run for weeks. His coach set up an arithmetic training plan, the same increase every week. The race registration system needs a completed training log as proof of preparation before it'll issue an entry code.
Every calculation gets him one step closer to the start line.
📋 Josh's Training Plan, Weekly Distance
First term a = 10 · Arithmetic sequence · Formula: tₙ = a + (n − 1) × d
1
Checkpoint 1, Common Difference
Looking at Josh's weekly distances: 10, 13, 16, 19, 22, 25, ...common difference (d) for this arithmetic sequence?
Subtract any term from the one after it, the answer should be the same every time.
Submit
Hint
💡 d = t₂ − t₁ = 13 − 10 = ?
✅ d = 3. Josh adds 3 km to his run every week, a classic arithmetic training plan.
2
Checkpoint 2, Find a Specific Term
Using a = 10 and d = 3 , how many kilometres does Josh run in week 8 ?
Use the formula: tₙ = a + (n − 1) × d. Substitute n = 8.
Submit
Hint
💡 t₈ = 10 + (8 − 1) × 3 = 10 + 7 × 3 = ?
✅ Week 8 = 31 km. The training is ramping up!
3
Checkpoint 3, Total Distance
What is the total distance Josh runs across the first 8 weeks of training?
Use the sum formula: Sₙ = n/2 × (a + last term). You know a = 10 and t₈ = 31 from the previous step.
Submit
Hint
💡 S₈ = 8/2 × (10 + 31) = 4 × 41 = ?
✅ 164 km total! That's some solid preparation over 8 weeks.
4
Checkpoint 4, Solve for n
The race distance is 52 km . In which week of the training plan does Josh first hit a training run of 52 km?
Set tₙ = 52 and solve for n: 10 + (n − 1) × 3 = 52. Rearrange step by step.
Submit
Hint
💡 10 + (n−1)×3 = 52 → (n−1)×3 = 42 → n−1 = 14 → n = ?
✅ Week 15! Josh hits the full race distance in week 15 of training.
5
Checkpoint 5, Work Backwards
A different athlete has an arithmetic plan where the 6th week is 35 km and the common difference is d = 5 .first-week distance (a)?
Use t₆ = a + (6−1) × 5 = 35. Solve for a.
Submit
Hint
💡 a + 5×5 = 35 → a + 25 = 35 → a = ?
✅ a = 10 km. They started at the same distance as Josh, same starting point, bigger weekly jump.
6
Checkpoint 6, Find d from Two Terms
A third athlete's training log shows their first week was 6 km and their 5th week was 22 km .common difference ?
From t₁ to t₅ is 4 gaps of d. Use: d = (t₅ − t₁) ÷ (5 − 1)
Submit
Hint
💡 d = (22 − 6) ÷ (5 − 1) = 16 ÷ 4 = ?
✅ d = 4. They increase by 4 km per week, faster progression than Josh's plan.
🏃
Entry Code Unlocked!
All 6 checkpoints verified. The training log is complete and the registration system has issued Josh's entry code for the Coastal Fun Run .
tₙ = a + (n−1)d · Sₙ = n/2 × (a + l) · d = (tₙ − t₁) ÷ (n − 1)
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