4
\begin{enumerate}[label=(\alph*)]
\item André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
\begin{enumerate}[label=(\roman*)]
\item How many counters are there in his sixth pile?
\item André makes ten piles of counters. How many counters has he used altogether?
\end{enumerate}\item In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start.

The probability $\mathrm { P } _ { n }$ of Betty starting on her $n$th throw is given by

$$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
\begin{enumerate}[label=(\roman*)]
\item Calculate $\mathrm { P } _ { 4 }$. Give your answer as a fraction.
\item The values $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots$ form an infinite geometric progression. State the first term and the common ratio of this progression.

Hence show that $\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1$.
\item Given that $\mathrm { P } _ { n } < 0.001$, show that $n$ satisfies the inequality

$$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$

Hence find the least value of $n$ for which $\mathrm { P } _ { n } < 0.001$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI C2  Q4 [12]}}