The diagram shows the curve with equation $y = \frac{1}{x}$. $n$ rectangles of equal width are d

Question

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. Let $n=2$. By considering $S_2$, show that $\frac{7}{12} < \ln 2$. State your reasoning clearly.[3] By considering $S_n$ for a suitable value of $n$, find a rational number $q$ such that $\frac{7}{12} < q < \ln 2$.[2] Express $S_n$ in the form $$\sum_{r=1}^n \mathrm{g}(r)$$ where the function $\mathrm{g}$ is to be determined.[1] 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]

Accepted Answer

(b)$\frac{37}{60}$ or $\frac{533}{840}$ or $\frac{1627}{2520}$ or $\frac{15797}{27720}$ (c)$\sum_{r=1}^n \frac{1}{n+r}$

2025 EJC P1 Q5

2025 EJC P1 Q5