Can we find the sum of the integers from $2k$ to $4k$ inclusive?

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1. The positive integer \(k\) is given.

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

   2. 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?

2. 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?