Sequence Difference Proof

For the sequence defined by $0, 2, 6, 12, 20, 30, 42$, show that $a_n - a_{n-1} = 2n$ and prove that $a_n = n(n+1)$ given $a_0 = 0$.

This problem involves understanding the structure and properties of a sequence, particularly focusing on the differences between consecutive terms and its closed form expression. When presented with a sequence, a common task in discrete mathematics is to identify patterns in the differences of consecutive terms. In this case, the sequence begins with simple quadratic differences, which suggest a polynomial relationship. To explore this, we start by examining the consecutive differences, which lead to the expression $2n$. This linear pattern in differences often suggests that the sequence itself might be represented by a quadratic formula.

Next, we move on to proving the closed form expression given for the sequence, which is $n(n+1)$. Proving such statements often involves methods like mathematical induction, where you demonstrate the base case and show that if a statement holds for an arbitrary case $n$, it also holds for $n+1$. This technique is foundational in discrete mathematics, allowing one to extend the truth of a statement across all natural numbers. Exploring the means of transition from the difference equation to the closed form also introduces the idea of telescoping sums, which is an important technique to understand cumulative effects over sequences.

Thus, this problem not only deals with deriving formulae for sequences but also reinforces understanding in proof strategies and recognizing patterns in sequences—a skill crucial for problem-solving in discrete mathematics.