2021 TMJC P1 Q7

A geometric progression has first term \(a\) and common ratio \(r\), and an arithmetic progression has first term \(b\) and common difference \(d\), where \(a\),\(b\),\(d\) and \(r\) are non-zero real numbers. The first, third and eighth term of the geometric progression are equal to the second, third and fifth term of the arithmetic progression respectively.

1. Show that \({{r}^{7}}-3{{r}^{2}}+2=0\).[2]
2. Find the values of \(r\), giving your answer correct to \(5\) decimal places. Hence explain why the sum to infinity of the geometric progression exists.[2]

It is now given that \(a=12\) and \(r\) is the positive value found in part (ii).

3. Let \(E\) be the sum of the first \(n\) even-numbered terms of the arithmetic progression and \({{S}_{\infty }}\) be the sum to infinity of the geometric progression. Find the largest value of \(n\) such that the difference between \(E\) and \({{S}_{\infty }}\) is less than \(1000\).[6]