Question 4 - A-Level Maths

Exam Board	OCR MEI
Module	C2 (Core Mathematics 2)
Topic	Arithmetic Sequences and Series

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.
1. How many counters are there in his sixth pile?
2. André makes ten piles of counters. How many counters has he used altogether?
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 $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
1. Calculate $\mathrm { P } _ { 4 }$. Give your answer as a fraction.
2. 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 $\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1$.
3. 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 $\mathrm { P } _ { n } < 0.001$.