Question 2
Study the following pattern: $$\frac1{1\times2}=\frac12,\quad \frac1{1\times2}+\frac1{2\times3}=\frac23,\quad \frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}=\frac34.$$ Given that $$\frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}+\cdots+\frac1{2013\times2014}=\frac{a+2}{a+3},$$ where $a$ is a positive integer. Find the value of $a$.