2023 NYJC MYE P1

The series \(a+ar-a{{r}^{2}}-a{{r}^{3}}+a{{r}^{4}}+a{{r}^{5}}-a{{r}^{6}}-a{{r}^{7}}+\)…, where \(a>0\), has its \(k\)th term, \({{T}_{k}}\), defined by \({{T}_{k}}=\left\{ \begin{matrix} a{{r}^{k-1}}, \\ -a{{r}^{k-1}}, \\ \end{matrix} \right.\) \(\begin{matrix} \mathrm{if}\,k=4p-3\,\,\mathrm{or}\,\,4p-2 \\ \mathrm{if}\,k=4p-1\,\,\mathrm{or}\,\,4p\,\,\,\,\,\,\, \\ \end{matrix}\), for \(p\in {{\mathbb{Z}}^{+}}\),

1. By rewriting the series as the sum of two geometric series, or otherwise, prove that the sum to \(4n\) terms of the series is \(\frac{a(1+r)(1-{{r}^{4n}})}{1+{{r}^{2}}}\).[3]
2. State the set of values of \(r\) for which the series has a sum to infinity.[1]
3. Given that \(r=-0.9\), find the least value of \(n\) for the sum of \(4n\) terms to be greater than \(99\%\) of the sum to infinity.[3]