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sum 1/1x4+1/4x7+1/7x10+1/1995x1996 , 1996 Chile NMO p7

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September 5, 2022
algebra

Problem Statement

(a) Let a,b,c,da, b, c, d be integers such that ad≠bcad\ne bc. Show that is always possible to write the fraction 1(ax+b)(cx+d)\frac{1}{(ax+b)(cx+d)}in the form rax+b+scx+d\frac{r}{ax+b}+\frac{s}{cx+d}
(b) Find the sum 11⋅4+14⋅7+17⋅10+...+11995⋅1996\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}
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