When the step slides back
A cycling tour rides 45 km on day one. Each day after is 3 km less: 45, 42, 39, 36, …
- Day 1: 45 km. Day 2: 42 km.
- The distances keep falling by 3 km a day.
- Day 16: 0 km. The tour has nothing left to ride.
- Day 17 would be −3 km. The story refuses it.
Rule: d is the step, and its sign, plus or minus, decides. A list grows when d is positive. It slides when d is negative. It stays flat when d is 0. A slide is just as real a progression as a climb.
The first 8 days cover 8/2 × (2 × 45 + 7 × (−3)) = 4 × 69 = 276 km. The slide ends on the day the distance reaches zero. 45 − 3(n − 1) = 0 gives 3(n − 1) = 45, so n = 16. Day 15 rides 3 km, the last distance above zero.
Up to day 16 the end pairing still works. Day 1 pairs with day 16: 45 + 0. Day 2 pairs with day 15: 42 + 3. The 16 days make 8 pairs of 45, which is 360 km.
Watch out!
A falling term does not make the total fall. The tour total keeps growing to 360 km. A term and a total are different things.
Slide the start and the step. Watch where the tour ends.
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