The positive integers are bracketed as follows: \[(1), (2,3), (4,5,6),\dotsc ,\] where there are \(r\) integers in the \(r^{th}\) bracket.

Find expressions for the first and last integers in the \(r^{th}\) bracket.

Can we find a pattern for the starting terms in the brackets? For the finishing terms?

Prove that the sum of the integers in the \(r^{th}\) bracket is \(\dfrac{1}{2}r(r^2+1)\).

Can we use the values for the first and the last integers in the \(r^{th}\) bracket from the first part?