2025 IGCSE Additional Mathematics October/November 0606/13 Q10

An arithmetic progression has first term \(a\) and common difference \(d\).

Given that \(S_{20}=3\times S_{10}\), find \(a\) in terms of \(d\).[3]

A geometric progression, A, has common ratio \(r\), where \(|r|<1\).

The terms of this progression are \(a_1,a_2,a_3,\ldots\).

Another geometric progression, B, has terms \(b_1,b_2,b_3,\ldots\), where

\(b_1=a_2,\quad b_2=a_4,\quad b_3=a_6,\ldots\).

The sum to infinity of A is \(S_A\) and the sum to infinity of B is \(S_B\).

Find \(\dfrac{S_B}{S_A}\) in terms of \(r\).

Give your answer in its simplest form.[5]