2007 H2 Mathematics

Paper 1 Question 10

A geometric series has a common ratio $r,$ and an arithmetic series has first term $a$ and common difference $d,$ where $a$ and $d$ are non-zero.

The first three terms of the geometric series are equal to the first, fourth and sixth terms respectively of the arithmetic series

(a)

Show that $3 r^{2} - 5 r + 2 .$

[4]

(b)

Deduce that the geometric series is convergent and find, in terms of $a,$ the sum to infinity.

[5]

(c)

The sum of the first $n$ terms of the arithmetic series is denoted by $S .$ Given that $a > 0,$ find the set of possible values of $n$ for which $S$ exceeds $4 a .$

[5]