2025 TMJC P1 Q4

An arithmetic series has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero real constants. A convergent geometric series has first term \(b\) and common ratio \(r\), where \(b > 0\) and \(r\) is a non-zero real constant. The sum of the first \(n\) terms of the geometric series is denoted by \(S\). It is given that the \(7\)th, \(10\)th and \(11\)th terms of the arithmetic series are equal to the \(3\)rd, \(6\)th and \(10\)th terms of the geometric series respectively.

1. Show that \(r\) satisfies the equation \(3r^7 - 4r^3 + 1 = 0\). Solve this equation, giving your answer correct to \(4\) decimal places.[4]
2. A new series is formed by taking the even numbered terms of the geometric series. Find the smallest value of \(n\) for which \(S\) differs from the sum to infinity of the new series by at least \(1.78b\).[4]