<p>A convergent geometric sequence of positive terms, $G$ has a non-zero first term $a$ and common ratio $r$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The sum of the first $n$ odd-numbered terms of $G$ is equal to the sum of all terms after the $\left( 2n-1 \right)$th term of $G$. Show that $2{{r}^{2n}}+{{r}^{2n-1}}-1=0$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">In another sequence $H$, each term is the reciprocal of the corresponding term of $G$. If the $n$th term of $G$ and $H$ is denoted by ${{u}_{n}}$ and ${{v}_{n}}$ respectively, show that a new sequence whose $n$th term is $\ln \left( \frac{{{u}_{n}}}{{{v}_{n}}} \right)$, is an arithmetic progression.</div><p style="text-align: right;">[2]</p></li></ol>

A convergent geometric sequence of positive terms, $G$ has a non-zero first term $a$ and common ratio $r$.

The sum of the first $n$ odd-numbered terms of $G$ is equal to the sum of all terms after the $\left( 2n-1 \right)$th term of $G$. Show that $2{{r}^{2n}}+{{r}^{2n-1}}-1=0$.

In another sequence $H$, each term is the reciprocal of the corresponding term of $G$. If the $n$th term of $G$ and $H$ is denoted by ${{u}_{n}}$ and ${{v}_{n}}$ respectively, show that a new sequence whose $n$th term is $\ln \left( \frac{{{u}_{n}}}{{{v}_{n}}} \right)$, is an arithmetic progression.

2018 SRJC P1 Q6

2018 SRJC P1 Q6

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