Question:

Step 1: To find the value of $m$, we start by examining the given expression $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}$. We notice that each term can be simplified using the difference of squares formula: $\dfrac{1}{\sqrt{k}+\sqrt{k+1}} = \dfrac{\sqrt{k+1}-\sqrt{k}}{(\sqrt{k}+\sqrt{k+1})(\sqrt{k+1}-\sqrt{k})} = \dfrac{\sqrt{k+1}-\sqrt{k}}{(k+1)-k} = \sqrt{k+1}-\sqrt{k}$. Step 2: Applying the simplification from Step 1 to each term in the sum, we get: $m = (\sqrt{2}-\sqrt{1}) + (\sqrt{3}-\sqrt{2}) + \cdots + (\sqrt{100}-\sqrt{99})$. This is a telescoping series where all intermediate terms cancel out, leaving us with $m = \sqrt{100} - \sqrt{1} = 10 - 1 = 9$. Step 3: Next, we evaluate the expression for $n$: $\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\cdots+\dfrac{1}{99\cdot100}$. Each term in this sum can be rewritten as $\dfrac{1}{k(k+1)} = \dfrac{1}{k} - \dfrac{1}{k+1}$. Thus, the sum becomes $n = \left(1 - \dfrac{1}{2}\right) + \left(\dfrac{1}{2} - \dfrac{1}{3}\right) + \cdots + \left(\dfrac{1}{99} - \dfrac{1}{100}\right)$. Step 4: Similar to the series for $m$, the series for $n$ is also telescoping. All terms except the first and the last cancel out, yielding $n = 1 - \dfrac{1}{100} = \dfrac{100-1}{100} = \dfrac{99}{100}$. Step 5: Now that we have $m = 9$ and $n = \dfrac{99}{100}$, we can determine the equation of the line on which the point $(m, n)$ lies. Given the options, we need to find which line passes through the point $(9, \dfrac{99}{100})$. To do this, we can substitute $m$ and $n$ into each of the given equations and see which one holds true. Step 6: Let's examine Option 2: $11x - 100y = 0$. Substituting $x = m = 9$ and $y = n = \dfrac{99}{100}$ into the equation gives $11(9) - 100\left(\dfrac{99}{100}\right) = 99 - 99 = 0$. This confirms that the point $(9, \dfrac{99}{100})$ lies on the line $11x - 100y = 0$. Step 7: Since the point $(m, n)$ satisfies the equation $11x - 100y = 0$, the final answer is that the point $(m, n)$ lies on the line given by Option 2. Therefore, the correct equation is $11x - 100y = 0$, and the final answer is $\boxed{11x-100y=0}$, which corresponds to Option 2. The final answer is $\boxed{2}$.