Sequence and Series - Arithmetic Progression

Clara organises cans in triangular piles, where each row has one less can than the row below. For example, the pile of \(15\) cans shown has \(5\) cans in the bottom row and \(4\) cans in the row above it.

1. A pile has \(20\) cans in the bottom row. Show that the pile contains \(210\) cans.[4]
2. There are \(3240\) cans in a pile. How many cans are in the bottom row?[4]
3. 1. There are \(S\) cans and they are organised in a triangle pile with \(n\) cans in the bottom row. Show \(n^2 + n - 2S = 0\).
   2. Clara has \(2100\) cans. Explain why she cannot organise them in a triangular pile.[6]