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<ol class="abc-list"><li><div class="list-item-paragraph">A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{u}_{n}}=\frac{1}{4}\left( 7-{{3}^{n}} \right)$, for $n\ge 1$.</div><ol class="i-list"><li><div class="list-item-paragraph">By expressing $4{{u}_{n+1}}$ and $12{{u}_{n}}$ in terms of $n$, or otherwise, find a recurrence relation of the form</div><div class="list-item-paragraph" style="text-align: center">${{u}_{n+1}}=\mathrm{f}\left( {{u}_{n}} \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find $\sum\limits_{r=2}^{n}{{{u}_{r}}}$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by ${{S}_{n}}=2-\frac{1}{\left( n+1 \right)!}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find a formula for ${{v}_{n}}$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Explain why the series $\sum\limits_{r=1}^{\infty }{{{v}_{r}}}$ converges. Hence, state the value of the sum to infinity.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

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