2025 SAJC Promo Q4

An arithmetic series has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. A convergent geometric series has first term \(b\) and common ratio \(r\), where \(b\) is positive and \(r\) is non-zero.

It is given that the \(1^{\text{st}}\) term of the arithmetic series is three times the \(6^{\text{th}}\) term of the geometric series and the \(6^{\text{th}}\) term of the arithmetic series is equal to the \(4^{\text{th}}\) term of the geometric series.

It is also given that the sum of the \(11^{\text{th}}\) term of the arithmetic series and the \(6^{\text{th}}\) term of the geometric series is equal to the sum of the \(4^{\text{th}}\) and \(5^{\text{th}}\) term of the geometric series.

1. Show that \(r\) satisfies the equation \(2r^2+r-1=0\) and find the value of \(r\).[4]
2. Hence, find the smallest value of \(n\) for which the sum of the first \(n\) terms of the geometric series differs from the sum to infinity of the geometric series by less than \(0.3b\).[3]