Question 1178378
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For a minute, forget about the factor of  (1/3)  and work with the numerators only


    r*(r+1)*(r+2) - (r-1)*r*(r+1) = r*(r+1) * [ (r+2) - (r-1) ] = 3r*(r+1).



Now recall about this factor (1/3) and get


    f(r) - f(r-1) = r*(r+1)


in the simplest form.
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Solved.