Review question

# What's the sum of the $r^{th}$ bracket in $(1), (2,3), (4,5,6),\cdots$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7240

## Question

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 $\frac{1}{2} n (n+1)$.