SOLUTION: If An=1/n(n+1) for all positive integer n, What is the sum of the first 100 elements of An?

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Question 1057149: If An=1/n(n+1) for all positive integer n, What is the sum of the first 100 elements of An?
Answer by ikleyn(52851) About Me  (Show Source):
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If An=1/n(n+1) for all positive integer n, What is the sum of the first 100 elements of An?
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S%5Bn%5D = 1%2F2+%2B+1%2F6+%2B+1%2F12+%2B+ellipsis+%2B+1%2F%28n%2A%28n%2B1%29%29.   (1)


Each term 1%2F%28k%2A%28k%2B1%29%29 is the difference of two terms


1%2F%28k%2A%28k%2B1%29%29 = 1%2Fk+-+1%2F%28k%2B1%29.


If you replace each term in the sum (1) by the correcsponding difference and cancel all adjacent terms with opposite signs, you will get


S%5Bn%5D = 1 - 1%2F%28n%2B1%29.


At n = 100,  S%5B100%5D = 1+-+1%2F101.

Answer.  S%5B100%5D%5D = 1+-+1%2F101.