Câu hỏi 1
Cho $S_1-S_2+S_3-S_4+S_5=\frac{m}{n}$, trong đó $m$ và $n$ là các số nguyên dương nguyên tố cùng nhau, và
$$\begin{aligned}S_1&=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}.\end{aligned}$$
$$\begin{aligned}S_2&=\frac{1}{2\times3}+\frac{1}{2\times4}+\frac{1}{2\times5}+\frac{1}{2\times6}+\frac{1}{3\times4}\\ &\quad+\frac{1}{3\times5}+\frac{1}{3\times6}+\frac{1}{4\times5}+\frac{1}{4\times6}+\frac{1}{5\times6}.\end{aligned}$$
$$\begin{aligned}S_3&=\frac{1}{2\times3\times4}+\frac{1}{2\times3\times5}+\frac{1}{2\times3\times6}+\frac{1}{2\times4\times5}+\frac{1}{2\times4\times6}\\ &\quad+\frac{1}{2\times5\times6}+\frac{1}{3\times4\times5}+\frac{1}{3\times4\times6}+\frac{1}{3\times5\times6}+\frac{1}{4\times5\times6}.\end{aligned}$$
$$\begin{aligned}S_4&=\frac{1}{2\times3\times4\times5}+\frac{1}{2\times3\times4\times6}+\frac{1}{2\times3\times5\times6}+\frac{1}{2\times4\times5\times6}+\frac{1}{3\times4\times5\times6}.\end{aligned}$$
$$\begin{aligned}S_5&=\frac{1}{2\times3\times4\times5\times6}.\end{aligned}$$
Hãy tìm giá trị của $m+n$.
$$\begin{aligned}S_1&=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}.\end{aligned}$$
$$\begin{aligned}S_2&=\frac{1}{2\times3}+\frac{1}{2\times4}+\frac{1}{2\times5}+\frac{1}{2\times6}+\frac{1}{3\times4}\\ &\quad+\frac{1}{3\times5}+\frac{1}{3\times6}+\frac{1}{4\times5}+\frac{1}{4\times6}+\frac{1}{5\times6}.\end{aligned}$$
$$\begin{aligned}S_3&=\frac{1}{2\times3\times4}+\frac{1}{2\times3\times5}+\frac{1}{2\times3\times6}+\frac{1}{2\times4\times5}+\frac{1}{2\times4\times6}\\ &\quad+\frac{1}{2\times5\times6}+\frac{1}{3\times4\times5}+\frac{1}{3\times4\times6}+\frac{1}{3\times5\times6}+\frac{1}{4\times5\times6}.\end{aligned}$$
$$\begin{aligned}S_4&=\frac{1}{2\times3\times4\times5}+\frac{1}{2\times3\times4\times6}+\frac{1}{2\times3\times5\times6}+\frac{1}{2\times4\times5\times6}+\frac{1}{3\times4\times5\times6}.\end{aligned}$$
$$\begin{aligned}S_5&=\frac{1}{2\times3\times4\times5\times6}.\end{aligned}$$
Hãy tìm giá trị của $m+n$.