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Estonian Math Competitions

Algebra Difficulty 4.7 AIME Find the answer Estonia

Find the value of
$\frac{1 \cdot 3}{3 \cdot 5} + \frac{2 \cdot 4}{5 \cdot 7} + \frac{3 \cdot 5}{7 \cdot 9} + \frac{4 \cdot 6}{9 \cdot 11} + \dots + \frac{1009 \cdot 1011}{2019 \cdot 2021}$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Answer: $\frac{505 \cdot 1009}{2021} = \frac{509545}{2021} = 252 + \frac{253}{2021}$ .

The sum consists of $1009$ terms, where the $i$ -th term is of the form $\frac{i(i+2)}{(2i+1)(2i+3)}$ . Let $s$ be the desired sum. Notice that
$\frac{i(i+2)}{(2i+1)(2i+3)} = \frac{1}{4} - \frac{3}{4} \cdot \frac{1}{(2i+1)(2i+3)} = \frac{1}{4} - \frac{3}{8} \cdot \left( \frac{1}{2i+1} - \frac{1}{2i+3} \right)$
Therefore
$\begin{aligned} s &= 1009 \cdot \frac{1}{4} - \frac{3}{8} \cdot \left( \left( \frac{1}{3} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{7} \right) + \dots + \left( \frac{1}{2019} - \frac{1}{2021} \right) \right) \\ &= \frac{1009}{4} - \frac{3}{8} \cdot \left( \frac{1}{3} - \frac{1}{2021} \right) = \frac{1009}{4} - \frac{3}{8} \cdot \frac{2021-3}{3 \cdot 2021} = \frac{1009}{4} - \frac{2018}{8 \cdot 2021} \\ &= \frac{1009}{4} - \frac{1009}{4 \cdot 2021} = \frac{1009}{4} \cdot \left( 1 - \frac{1}{2021} \right) = \frac{1009}{4} \cdot \frac{2020}{2021} = \frac{505 \cdot 1009}{2021} = \frac{509545}{2021} \end{aligned}$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.