The non-zero numbers $a$, $b$ and $c$ are the first, third and fifth terms of an arithmetic series respectively. Write down an expression for $b$ in terms of $a$ and $c$. [1] Write down an expression for the common difference, $d$, of this arithmetic series in terms of $a$ and $b$. [1] Hence show that the sum of the first ten terms can be expressed as $\frac{5}{4}\left( 9c-a \right)$. [2] If $a$, $b$ and $c$ are also the fourth, third and first terms, respectively of a geometric series, find the common ratio of this series in terms of $a$ and $b$ and hence show that $\left( 2b-a \right){{a}^{2}}={{b}^{3}}$ The $n$th term of a geometric series is ${{\left( 2{{\sin }^{2}}\alpha \right)}^{n-1}}$. Find all the values of $\alpha$, where $0\le \alpha \le 2\pi $, such that the series is convergent. For the values of $\alpha$ found in part (i), find the sum to infinity, simplifying your answer.

2019 RI P2 Q4

4 marks

Prelims

The non-zero numbers $a$, $b$ and $c$ are the first, third and fifth terms of an arithmetic series respectively.
1. Write down an expression for $b$ in terms of $a$ and $c$.[1]
2. Write down an expression for the common difference, $d$, of this arithmetic series in terms of $a$ and $b$.[1]
3. Hence show that the sum of the first ten terms can be expressed as[2]
  $\frac{5}{4}\left( 9c-a \right)$.[2]
4. If $a$, $b$ and $c$ are also the fourth, third and first terms, respectively of a geometric series, find the common ratio of this series in terms of $a$ and $b$ and hence show that
  $\left( 2b-a \right){{a}^{2}}={{b}^{3}}$
The $n$th term of a geometric series is ${{\left( 2{{\sin }^{2}}\alpha \right)}^{n-1}}$.
1. Find all the values of $\alpha$, where $0\le \alpha \le 2\pi $, such that the series is convergent.
2. For the values of $\alpha$ found in part (i), find the sum to infinity, simplifying your answer.

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