N2016 P1 Q4

An arithmetic series has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. A geometric series has first term \(b\) and common ratio \(r\), where \(b\) and \(r\) are non-zero. It is given that the 4th, 9th and 12th terms of the arithmetic series are equal to the 5th, 8th and 15th terms of the geometric series respectively.

1. Show that \(r\) satisfies the equation \(5r^{10}-8r^3+3=0\). Given that \(|r|<1\), solve this equation, giving your answer correct to \(2\) decimal places.[4]
2. Using this value of \(r\), find, in terms of \(b\) and \(n\), the sum of the terms of the geometric series after, but not including, the \(n\)th term, simplifying your answer.[3]