<p>The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}},{{u}_{2}},{{u}_{3}},...$ is given by</p>
<p style="text-align: center">${{S}_{n}}=k{{n}^{2}}+\left( 5-k \right)n$,</p>
<p>where $k$ is an unknown constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find ${{u}_{n}}$ in terms of $n$ and $k$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence show that the sequence is an arithmetic progression.</span></div><p style="text-align: right;">[1]</p></li></ol>
<p>The ${{r}^{\mathrm{th}}}$ term, ${{v}_{r}}$, of another sequence ${{v}_{1}},{{v}_{2}},{{v}_{3}},...,$ is given by ${{v}_{r}}={{\mathrm{e}}^{{{u}_{r}}}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Show that this sequence is geometric and determine the values of $k$ for which the sum to infinity exists.</span></div><p style="text-align: right;">[2]</p></li></ol>

The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}},{{u}_{2}},{{u}_{3}},...$ is given by

${{S}_{n}}=k{{n}^{2}}+\left( 5-k \right)n$,

Find ${{u}_{n}}$ in terms of $n$ and $k$.

Hence show that the sequence is an arithmetic progression.

The ${{r}^{\mathrm{th}}}$ term, ${{v}_{r}}$, of another sequence ${{v}_{1}},{{v}_{2}},{{v}_{3}},...,$ is given by ${{v}_{r}}={{\mathrm{e}}^{{{u}_{r}}}}$.

Show that this sequence is geometric and determine the values of $k$ for which the sum to infinity exists.

2019 ACJC Promo Q4

2019 ACJC Promo Q4

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