2025 EJC P1 Q5

The diagram shows the curve with equation \(y = \frac{1}{x}\).
\(n\) rectangles of equal width are drawn under the curve between \(x=1\) and \(x=2\).

Let \(S_n\) be the total area of the \(n\) rectangles.

1. Let \(n=2\). By considering \(S_2\), show that \(\frac{7}{12} < \ln 2\). State your reasoning clearly.[3]
2. By considering \(S_n\) for a suitable value of \(n\), find a rational number \(q\) such that \(\frac{7}{12} < q < \ln 2\).[2]

3. Express \(S_n\) in the for

   \(\sum_{r=1}^n \mathrm{g}(r)\),

   where the function \(\mathrm{g}\) is to be determined.

   [1]
4. By considering another suitable set of \(n\) rectangles, show with the aid of a diagram that \[\ln 2 < S_n + \frac{1}{2n}\] where \(n\) is any given positive integer.[3]