The positive integer \(k\) is given.

Find, in terms of \(k\), an expression for \(S_1\), the sum of the integers from \(2k\) to \(4k\) inclusive.

Find, in terms of \(k\), an expression for \(S_2\), the sum of the

*odd*integers lying between \(2k\) and \(4k\).Show that \(\dfrac{S_1}{S_2} = 2+\dfrac{1}{k}\).

Prove that the sum of the first \(n\) terms of the geometric progression having first term \(a\) and common ratio \(r\) (\(r \neq 1\)) is \[a \left( \frac{1-r^n}{1-r} \right).\] By regarding the recurring decimal \(0.\dot{0}7\dot{5}\) (\(=0.075075\dotsc\), where the figures \(075\) repeat) as an infinite geometric progression, or otherwise, obtain the value of the decimal as a fraction in its lowest terms.