Challenging +1.2 This question requires setting up simultaneous equations from the given conditions, solving a quadratic to find the common ratio, and computing a sum to infinity. While it involves multiple steps and algebraic manipulation with surds, the techniques are standard A-level: using GP/AP formulas, solving quadratics, and applying the convergence condition |r|<1. The 12 marks reflect length rather than conceptual difficulty—it's a solid multi-step problem but follows predictable patterns without requiring novel insight.
In this question you must show detailed reasoning.
The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\).
Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]
In this question you must show detailed reasoning.
The $n$th term of a geometric progression is denoted by $g_n$ and the $n$th term of an arithmetic progression is denoted by $a_n$. It is given that $g_1 = a_1 = 1 + \sqrt{5}$, $g_2 = a_2$ and $g_3 + a_3 = 0$.
Given also that the geometric progression is convergent, show that its sum to infinity is $4 + 2\sqrt{5}$. [12]
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q8 [12]}}