Sum of Arithmetic Sequence Proof

Show that $1 + 2 + \ldots + n = \frac{n(n + 1)}{2}$.

The given problem asks you to prove a well-known formula, which is the sum of the first n natural numbers. This is a classic problem that helps illustrate the principles of mathematical induction, a powerful proof technique often used in discrete mathematics. Induction involves two major steps: the base case and the inductive step. In this context, the base case typically involves proving the statement for the initial value of your sequence, usually n equals 1. Once the base case is verified, the inductive step requires you to assume the statement holds for some arbitrary natural number k, and subsequently prove it for k plus 1.

The very essence of induction mirrors the idea of climbing a ladder: proving the base case establishes the first rung, and the inductive step ensures that if you can reach one rung, you can reach the next. This problem also touches upon the deeper understanding of arithmetic sequences. An arithmetic sequence has a constant difference between consecutive terms. Here, the sum of the first n terms of this sequence adds up smoothly to a quadratic expression, intriguing in its own right.

One way to gain insight is to consider pairing terms from opposite ends of the sequence, which consistently sum to the same intermediate value, thereby simplifying the summation. Approaching problems with this mindset not only helps in proofs but also enhances problem-solving skills in tackling sequences and series in other mathematical contexts.