Rabbit & Turtle Race

Once upon a time, a rabbit and a turtle were competing in a race.

The fast rabbit could hop in an increasing distance similar to the Fibonacci sequence (omitting the first 1-term) as shown above: \(1, 2, 3, 5, 8, 13,\ldots.\)

The slow turtle could roll in an increasing distance of an arithmetic sequence of 1-interval as shown: \(1, 2, 3, 4, 5,\ldots\)

Though seemingly even at the first three steps, soon afterwards, the rabbit rapidly went ahead of his opponent. However, at one point, the rabbit, confident of his victory, stopped for a nap. Later on, the turtle continued his track in the same pattern and met the rabbit at that same distance. The turtle then carried on his effort before eventually winning the race.

According to this tale, what is the least possible distance from the start to the rabbit's sleeping point?


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