<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$,${{u}_{3}}$, … is defined by</p>
<p style="text-align: center">${{u}_{n}}=\sum\limits_{r=1}^{n}{\left( 2r+n+1 \right)}$.</p>
<p>Another sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by ${{v}_{n}}=\frac{2}{{{u}_{n}}}$, where&nbsp;$n\in {{\mathbb{Z}}^{+}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{u}_{n}}$ in terms of $n$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;${{v}_{n}}=\frac{1}{n}-\frac{1}{n+1}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Describe the behaviour of the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$… .<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the sum, ${{S}_{N}}$, of the first $N$ terms of the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$… .<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Give a reason why the series ${{S}_{N}}$ converges, and write down the value of the sum to infinity.<span class="inline-marks">[1]</span></div></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$,${{u}_{3}}$, … is defined by

${{u}_{n}}=\sum\limits_{r=1}^{n}{\left( 2r+n+1 \right)}$.

Another sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by ${{v}_{n}}=\frac{2}{{{u}_{n}}}$, where $n\in {{\mathbb{Z}}^{+}}$.

Show that ${{v}_{n}}=\frac{1}{n}-\frac{1}{n+1}$.

Describe the behaviour of the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$… .

Find the sum, ${{S}_{N}}$, of the first $N$ terms of the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$… .

Give a reason why the series ${{S}_{N}}$ converges, and write down the value of the sum to infinity.

2021 RI Promo Q4

2021 RI Promo Q4

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