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1/1 \cdot 4 + 1/4\cdot 7 + 1/7\cdot 10+ ...+1/2998 \cdot 3001, sum with hint

Source: Brazilian Mathematical Olympiad 1985 P1

July 27, 2018
algebraSum

Problem Statement

a,b,c,da, b, c, d are integers with ad≠bcad \ne bc. Show that 1/((ax+b)(cx+d))1/((ax+b)(cx+d)) can be written in the form r/(ax+b)+s/(cx+d) r/(ax+b) + s/(cx+d). Find the sum 1/1⋅4+1/4⋅7+1/7⋅10+...+1/2998⋅30011/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001.
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