Explanation vs Derivation Math Example 2

Solution

1. 1
   Derivation (induction): Base $n=1$: $1 = \frac{1 \cdot 2}{2}$. Inductive step: $\sum_{k=1}^{n+1}k = \frac{n(n+1)}{2}+(n+1) = \frac{(n+1)(n+2)}{2}$. QED.
2. 2
   Explanation (Gauss's pairing): Write the sum forwards and backwards: $S = 1+2+\cdots+n$ and $S = n+(n-1)+\cdots+1$. Adding: $2S = n$ pairs each summing to $n+1$. So $S = \frac{n(n+1)}{2}$.
3. 3
   The induction proves it; the pairing explains why — the symmetry of pairing first-and-last terms is the geometric insight.