Question

Докажите,что при любом n принадлежащем N выполняется равенство

Answer 1

$\dfrac{1}{5\cdot 12}+\dfrac{1}{12\cdot19}+...+\dfrac{1}{(7n-2)(7n+5)}=\\ \\ \\ =\dfrac{1}{7}\left(\dfrac{12-5}{5\cdot 12}+\dfrac{19-12}{12\cdot19}+...+\dfrac{7n+5-(7n-2)}{(7n-2)(7n+5)}\right)=\\ \\ \\ =\dfrac{1}{7}\left(\dfrac{1}{5}-\dfrac{1}{12}+\dfrac{1}{12}-\dfrac{1}{19}+...+\dfrac{1}{7n-2}-\dfrac{1}{7n+5}\right)=\\ \\ \\ =\dfrac{1}{7}\left(\dfrac{1}{5}-\dfrac{1}{7n+5}\right)=\dfrac{1}{7}\cdot\dfrac{7n+5-5}{5(7n+5)}=\dfrac{n}{5(7n+5)},~~\forall\,\,\,n \in \mathbb{N}$