<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that $\mathrm{f}$ is a continuous and increasing function, explain, with an aid of a sketch, why the value of</span></div><div class="list-item-paragraph" style="text-align: center">$\underset{n\to \infty }{\mathop{\lim }}\,\frac{2}{n}\left[ \mathrm{f}\left( 1 \right)+\mathrm{f}\left( 1+\frac{2}{n} \right)+\mathrm{f}\left( 1+\frac{4}{n} \right)+...+\mathrm{f}\left( 1+\frac{2\left( n-1 \right)}{n} \right) \right]$</div><div class="list-item-paragraph">is $\int_{1}^{3}{\mathrm{f}\left( x \right)}\,\mathrm{d}x$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, evaluate $\underset{n\to \infty }{\mathop{\lim }}\,\frac{2}{n}\left( {{\mathrm{e}}^{2}}+{{\mathrm{e}}^{2+\frac{4}{n}}}+{{\mathrm{e}}^{2+\frac{8}{n}}}+...+{{\mathrm{e}}^{2+\frac{4\left( n-1 \right)}{n}}} \right)$, leaving your answer in exact form.</span></div><p style="text-align: right;">[2]</p></li></ol>

Given that $\mathrm{f}$ is a continuous and increasing function, explain, with an aid of a sketch, why the value of

$\underset{n\to \infty }{\mathop{\lim }}\,\frac{2}{n}\left[ \mathrm{f}\left( 1 \right)+\mathrm{f}\left( 1+\frac{2}{n} \right)+\mathrm{f}\left( 1+\frac{4}{n} \right)+...+\mathrm{f}\left( 1+\frac{2\left( n-1 \right)}{n} \right) \right]$

is $\displaystyle\displaystyle \int_{1}^{3}{\mathrm{f}\left( x \right)}\,\mathrm{d}x$.

Hence, evaluate $\underset{n\to \infty }{\mathop{\lim }}\,\frac{2}{n}\left( {{\mathrm{e}}^{2}}+{{\mathrm{e}}^{2+\frac{4}{n}}}+{{\mathrm{e}}^{2+\frac{8}{n}}}+...+{{\mathrm{e}}^{2+\frac{4\left( n-1 \right)}{n}}} \right)$, leaving your answer in exact form.

2022 DHS Promo Q2

2022 DHS Promo Q2

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