Question 1057149
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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[n]}}} = {{{1/2 + 1/6 + 1/12 + ellipsis + 1/(n*(n+1))}}}.   (1)


Each term {{{1/(k*(k+1))}}} is the difference of two terms


{{{1/(k*(k+1))}}} = {{{1/k - 1/(k+1)}}}.


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[n]}}} = 1 - {{{1/(n+1)}}}.


At n = 100,  {{{S[100]}}} = {{{1 - 1/101}}}.

<U>Answer</U>.  {{{S[100]]}}} = {{{1 - 1/101}}}.
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