Olympiad workbookIOQMSequences and Series

Let the sum _ n=1 ^9 1 n(n+1)(n+2) written in its lowest terms be p q . Find the value of q-p .

Correct answer

Step-by-step solution

aligned & _ n=1 ^9 1 2 n+2-n n(n+1)(n+2) & = 1 2 _ n=1 ^9 ( 1 n(n+1) - 1 (n+1)(n+2) ) & = 1 2 ( ( 1 1.2 - 1 2.3 )+ ( 1 2.3 - 1 3.4 )+ + ( 1 9 10 - 1 10 11 ) ) & = 1 2 ( 1 2 - 1 110 ) & = 1 2 ( 55-1 110 ) 27 110 q-p=110-27=83 aligned

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