Summation simplification help

Question

Hi, this is an answer to a sum simplication. However, I understand the highlighted. How did the sum all of a sudden change from index 3 to index 1? I do realize that in line 2, the difference in summation is equivalent to the Left side hand side.

If anyone can explain what happens in Lines 1 and 2 can help me understand.

Answer 1

The following mimics what happens (be not distracted by the summands):

For $n \geq 3$ we have $$ \sum_{k=3}^{n}k = \sum_{k=3}^{n}k + \bigg( \sum_{k=1}^{2}k - \sum_{k=1}^{2}k \bigg) = \bigg( \sum_{k=3}^{n}k + \sum_{k=1}^{2}k \bigg) - \sum_{k=1}^{2}k = \sum_{k=1}^{n}k - \sum_{k=1}^{2}k $$