# Series and Annuities: The Tortoise and the Hare Series

You may have heard the fable about the tortoise and the hare. Suppose that the tortoise and the hare are running a 5000 M race. The tortoise proceeds very slowly, never changing its speed. The hare runs very quickly at the start. The tortoise travels 10 M every minute. the hare travels 2500 M in the first minute but, in each minute after that,travels only half of the remaining distance (in the second minute, the hare travels half of the remaining 2500 M and so on).

A) Write the series for the distance the tortoise travels for the first few minutes of the race, where Sn represents the total distance traveled after n minutes. Each term of the series will represent the distance the tortoise travels in the minute.

B) Write the series for the distance the hare travels for the first few minutes of the race, where Sn represents the total distance traveled after n minutes. Each term of the series will represent the distance the hare travels in that minute.

C) Analyze and compare the two series. Include information such as Sn for both the tortoise and the hare, and your conclusion as who will win the race. Provide support for your analysis.

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#### Solution Preview

A) Sn: 10, 20, 30, 40, ?.. 10*n

B) Sn: 2500, (2500 + 2500/2), (2500 + 2500/2 + 2500/2^2 ), (2500 + 2500/2 + 2500/2^2 + 2500/2^3 ), ?? (2500 + 2500/2 + 2500/2^2 +? 2500/2^n-1)

C) A) Is an arithmetic sequence whose common difference is 10.

B) Is a sequence where the difference of ...