<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">A non-zero sequence $\{{{u}_{n}}\}$ where $n\ge 1$ and ${{u}_{i}}\ne {{u}_{j}}$ when $i\ne j$ is such that $\frac{1}{{{u}_{n+1}}-{{u}_{n}}}$, $\frac{1}{2{{u}_{n+1}}}$, $\frac{1}{{{u}_{n+1}}-{{u}_{n+2}}}$ are consecutive terms of an arithmetic progression. Show that $\{{{u}_{n}}\}$ forms a geometric progression.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">A geometric series has first term $1$ and common ratio $r$, where $0 < r < 1$. The sum of the first $12$ terms is twice the sum of the terms from the $13$th term to the $36$th term (both inclusive). Show that ${{r}^{12}}=\frac{1}{2}\left( \sqrt{3}-1 \right)$.</span></div><p style="text-align: right;">[5]</p></li></ol>

A non-zero sequence $\{{{u}_{n}}\}$ where $n\ge 1$ and ${{u}_{i}}\ne {{u}_{j}}$ when $i\ne j$ is such that $\frac{1}{{{u}_{n+1}}-{{u}_{n}}}$, $\frac{1}{2{{u}_{n+1}}}$, $\frac{1}{{{u}_{n+1}}-{{u}_{n+2}}}$ are consecutive terms of an arithmetic progression. Show that $\{{{u}_{n}}\}$ forms a geometric progression.

A geometric series has first term $1$ and common ratio $r$, where $0 < r < 1$. The sum of the first $12$ terms is twice the sum of the terms from the $13$th term to the $36$th term (both inclusive). Show that ${{r}^{12}}=\frac{1}{2}\left( \sqrt{3}-1 \right)$.

2022 HCI J1 BT Q5

2022 HCI J1 BT Q5

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