中学受験によく出る部分分数分解の問題と解き方

$\ \ \ \ \dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}\\ =\dfrac{1}{1 \times 2}+\dfrac{1}{2 \times 3}+\dfrac{1}{3 \times 4}+\dfrac{1}{4 \times 5}\\ =\left(\dfrac{1}{1}-\dfrac{1}{2}\right) +\left(\dfrac{1}{2}-\dfrac{1}{3}\right)\\ \ \ \ \ +\left(\dfrac{1}{3}-\dfrac{1}{4}\right) +\left(\dfrac{1}{4}-\dfrac{1}{5}\right)\\ =\dfrac{1}{1}-\dfrac{1}{5}\\ =\dfrac{4}{5}$

部分分数分解の公式

$\dfrac{1}{x(x+1)}=\dfrac{1}{x}-\dfrac{1}{x+1}$

$\dfrac{1}{2}=\dfrac{1}{1}-\dfrac{1}{2}$

$\dfrac{1}{6}=\dfrac{1}{2}-\dfrac{1}{3}$

$\dfrac{1}{12}=\dfrac{1}{3}-\dfrac{1}{4}$

$\dfrac{1}{20}=\dfrac{1}{4}-\dfrac{1}{5}$

$\dfrac{1}{30}=\dfrac{1}{5}-\dfrac{1}{6}$

$\dfrac{1}{42}=\dfrac{1}{6}-\dfrac{1}{7}$

$\dfrac{1}{56}=\dfrac{1}{7}-\dfrac{1}{8}$

部分分数分解の問題

$\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}\\ +\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}$

解答

$\ \ \ \ \dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}\\ =\dfrac{1}{1 \times 2}+\dfrac{1}{2 \times 3}+\dfrac{1}{3 \times 4}\\ \ \ \ \ +\dfrac{1}{4 \times 5}+\dfrac{1}{5 \times 6}+\dfrac{1}{6 \times 7}\\ =\left(\dfrac{1}{1}-\dfrac{1}{2}\right) +\left(\dfrac{1}{2}-\dfrac{1}{3}\right)\\ \ \ \ \ +\left(\dfrac{1}{3}-\dfrac{1}{4}\right) +\left(\dfrac{1}{4}-\dfrac{1}{5}\right)\\ \ \ \ \ +\left(\dfrac{1}{5}-\dfrac{1}{6}\right) +\left(\dfrac{1}{6}-\dfrac{1}{7}\right)\\ =\dfrac{1}{1}-\dfrac{1}{7}\\ =\dfrac{6}{7}$