2008 NJC P1 Q3
The graph of $y={{2}^{x}}$ for $0\le x\le 1$, is shown in the diagram below. Rectangles, each of width $\frac{1}{n}$, where $n$ is an integer are drawn under the curve.
Show that the total area of all the $n$ rectangles is $\frac{1}{n\left( {{2}^{\frac{1}{n}}}-1 \right)}$.
[2]
By considering the area of the region bounded by the curve $y={{2}^{x}}$, $x=1$ and the axes, briefly explain why $\ln 2<n\left( {{2}^{\frac{1}{n}}}-1 \right)$.
[3]
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