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 \(4k\).

Is there a convenient way of dividing up these sums into smaller sums that are easier to deal with?

- 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 \[a \left( \frac{1-r^n}{1-r} \right).\]

Can we use the fact that the \(k\)th term in this geometric progression has the form \(a\times r^{k-1}\)?

If we call the sum of the first n terms \(T_n\), do you notice anything about the terms in \(r\times T_n\)?

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.

What’s the first term of the geometric progression? What’s the common ratio?