Question 1022931
a) the tortoise series is an arithmetic series and
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Sn = 20 + 40 + 60 + .....+ n = (n/2)(a1 + an) where n is the number of terms to sum and a1 is the first term and an is the nth term
:
each term, an = a1 + d(n-1), where a1 is the first term, d is the common difference, and n is the nth term
:
d = 20
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b) the hare series is a geometric series and
:
Sn = 1000 + 500 + 250 + ....... + n = a((1 - r^n) / (1 - r)), where a is the first term and r is the common ratio
:
each term, an = ar^(n-1), where a is the first term and n is the nth term
:
r = (1/2)
:
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c) analysis 
: for the tortoise, 2000 = (n/2)(20 + an), and tortoise crosses the finish line in 14 minutes, that is, n = 14
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for the hare, the common ratio is (1/2) < 1
:
we know that the limit as n ----> infinity of a geometric sum is
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a / (1 - r) = 1000 / (1/2) = 2000
:
the hare will approach the finish line but never cross it   :-)
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