2025 SAJC P1 Q5

An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The \(3^{\mathrm{rd}}\), \(9^{\mathrm{th}}\) and \(11^{\mathrm{th}}\) terms of the arithmetic progression are three distinct consecutive terms of a geometric progression.

1. Find \(d\) in terms of \(a\).[3]

2. Let the sum of the first \(n\) terms of the arithmetic progression be \(S_n\), where \(S_{1} = 143\). Find the largest possible value of \(S_n\).

   [3]

The first term of the geometric progression is \(b\), where \(b\) is positive.

3. Determine whether the geometric series is convergent.[2]
4. Show that the sum of all terms after the \(n^{\mathrm{th}}\) term of the geometric progression is at most \(\frac{1}{2}b\), where \(n\) is a positive integer.[2]