Competition problem — Algebra — Maths Olympiad Prep

Algebra Difficulty 4.9 AIME Prove it Brazil

$a$ , $b$ , $c$ , $d$ are integers with $ad \neq bc$ . Show that $\frac{1}{(ax+b)(cx+d)}$ can be written in the form $\frac{r}{ax+b} + \frac{s}{cx+d}$ . Find the sum
$\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \dots + \frac{1}{2998 \cdot 3001}$

Solution

$\frac{a}{ax+b} - \frac{c}{cx+d} = \frac{ad-bc}{(ax+b)(cx+d)}, \text{ so we can take } r = \frac{a}{ad-bc}, s = -\frac{c}{ad-bc}.$

$\sum_{k=0}^{999} \frac{1}{(3k+1)(3k+4)} = \frac{1}{3} \sum_{k=0}^{999} \left( \frac{1}{3k+1} - \frac{1}{3k+4} \right) = \frac{1}{3} \left( 1 - \frac{1}{3001} \right) = \frac{1000}{3001}$

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