[tex] 1+2+3+4+...+12 = ? \\ \\\boxed{\text{Metoda I}} \\ \\ \text{Not\u{a}m }S = 1+2+3+4+...+12 \\ \text{Dar S este egal\u{a} \c{s}i cu: }12+11+10+9+...+1 \\ \\ \text{Deci:} \\ S = 1+2+3+4+....+12\\ S = 12+11+10+9+...+1 \\ --------------~(+) \\ S+S = (1+12)+(2+11)+(3+10)+(4+9)+...+(12+1) \\ 2S = \underset{\text{de 12 ori}}{\underbrace{13+13+13+13+...+13}} \\ \\ 2S = 13\cdot 12 \\ \\ S = \dfrac{13\cdot 12}{2} \\ \\ \Rightarrow S = 13\cdot 6 \Rightarrow \boxed{S = 78} [/tex]
[tex] \boxed{\text{Metoda 2}} \\ \\ \text{Aplic\u{a}m suma lui Gauss: } \boxed{1+2+3+4...+n = \dfrac{n\cdot (n+1)}{2}}\\ \\ \text{In cazul nostru: } \\ \\ 1+2+3+4+...+12 = \dfrac{12\cdot (12+1)}{2} = \dfrac{12\cdot 13}{2} = 6\cdot 13 = \boxed{78} [/tex]