**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}

\text{if}\,k=4p-3\,\,\text{or}\,\,4p-2 \\

\text{if}\,k=4p-1\,\,\text{or}\,\,4p\,\,\,\,\,\,\, \\

\end{matrix}$, for $p\in {{\mathbb{Z}}^{+}}$,

(a)

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]

(b)

State the set of values of $r$ for which the series has a sum to infinity.

[1]

(c)

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]

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