2022 RI P2 Q4

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 \(b\).[4]

\({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), \({{u}_{4}}\), …, \({{u}_{2n}}\) 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 \(n\).[4]

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.[5]