Question

Folosind metoda inductiei matematice, sa se demonstreze ca pentru orice numar natural nenul n, sunt adevarate egalitatile :

1*2+2*3+...+n(n+1)=n(n+1)(n+2) "supra" 3 =

Answer 1

Etapa verificarii .
P(1): 1(1+1)=1(1+1)(1+2)/3<=>2=2 (A)
Presupunem ca P(k) e adevarata si dem. ca P(k+1) e adv.
P(k): 1*2+2*3+...+k(k+1)=k(k+1)(k+2)/3
P(k+1):1*2+2*3+...k(k+1)+(k+1)(k+2)=(k+1)(k+2)(k+3)/3
k(k+1)(k+2)/3+(k+1)(k+2)=(k+1)(k+2)(k+3)/3 |*3
k(k+1)(k+2)+3(k+1)(k+2)=(k+1)(k+2)(k+3) | :(k+1)(k+2)
k+3=k+3 (A)
Conform principiului inductiei matematice ,P(n) e adevarata.