[-] Metoda I - Aducem formula la o expresie cunoscuta a fi adevarata.
Daca expandam suma :
2+4+6+8+....+n=n(n+1)
Daca impartim prin 2 in stanga si in dreapta egalului :
[tex]1+2+3+...+n=\frac{n(n+1)}{2}, \forall n\geq 1[/tex]
Am ajuns la formula pentru suma Gauss, formula care stim ca este adevarata.
[-] Metoda II - Inductie matematica
[tex]\text{Daca n = 1 :}\\\\ \sum_{k=1}^{1}(2k)=2=1(1+1)[/tex]
[tex]\text{Presupunem P(n) :}\\\sum_{k=1}^n2k=n(n+1) \text{ adevarata}, \forall n \in N, n\geq 1\\\\\text{Trebuie sa demonstram P(n+1) : }\\\sum_{k=1}^{n+1}2k = (n+1)(n+2)[/tex]
[tex]\sum_{k=1}^{n+1}2k \\\\=(\sum_{k=1}^{n}2k) + 2n \\\\=n(n+1) +2n\\\\=n^2+n+1+2n\\\\=n^2+3n+1\\\\=(n+1)(n+2) \text{, ceea ce trebuia demonstrat.}[/tex]
