What's the sum of the $r^{th}$ bracket in $(1), (2,3), (4,5,6),\cdots$?

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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.

Find the sum of all the integers in the first 20 brackets.

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

You may assume that the sum of the first \(n\) positive integers is \(\frac{1}{2} n (n+1)\).