<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The first $3$ terms of a geometric progression are $a$, $b$, $2$ and the first $3$ terms of an arithmetic progression are $2$, $a$, $b$, with non-zero common difference. Find the values of $a$ and $b$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, ${{u}_{4}}$, …, ${{u}_{2n}}$ is a sequence of $2n$ positive terms, with $n>1$. The odd-numbered terms form an arithmetic progression with common difference $p$ and the even-numbered terms form a geometric progression with common ratio $\frac{6}{5}$. Given that ${{u}_{1}}={{u}_{2}}=p$ and the sum of the odd-numbered terms is less than the sum of the even-numbered terms, find the least value of $n$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">A geometric progression of $n$ terms has first term $q$ and common ratio $r$, where $q$ is non-zero and $r\ne 1$. For $k\le n$, find the difference between the sum of the last $k$ terms and the sum of the first $k$ terms, simplifying your answer.</span></div><p style="text-align: right;">[5]</p></li></ol>

The first $3$ terms of a geometric progression are $a$, $b$, $2$ and the first $3$ terms of an arithmetic progression are $2$, $a$, $b$, with non-zero common difference. Find the values of $a$ and $b$.

${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, ${{u}_{4}}$, …, ${{u}_{2n}}$ is a sequence of $2n$ positive terms, with $n>1$. The odd-numbered terms form an arithmetic progression with common difference $p$ and the even-numbered terms form a geometric progression with common ratio $\frac{6}{5}$. Given that ${{u}_{1}}={{u}_{2}}=p$ and the sum of the odd-numbered terms is less than the sum of the even-numbered terms, find the least value of $n$.

A geometric progression of $n$ terms has first term $q$ and common ratio $r$, where $q$ is non-zero and $r\ne 1$. For $k\le n$, find the difference between the sum of the last $k$ terms and the sum of the first $k$ terms, simplifying your answer.

2022 RI P2 Q4

2022 RI P2 Q4

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