Question

1. Folosind metoda inductiei matematice sa se demonstreze ca pentru orice n apartine N' au loc egalitatile :
a) 1+2+3+...+ n = n(n+1) supra 2

2. Sa se demonstreze ca pentru n apartine N' au loc egalitatile :
a) 1 supra 1 ori 2 + 1 supra 3 ori 5 + ... + 1 supra n(n+1) = n supra n+1

Answer 1

a)
I. Etapa de verificare:
Pt n=1: 1=[1(1+1)]/2 <=> 1=2/2 <=> 1=1 (Adevarat)
Pt n=2: 1+2=[2(2+1)]/2 <=> 3=2*3/2 <=> 3=6/2 <=>3=3 (adevarat)
II. Etapa de demonstratie:
Presupunem P(k) adevarat
P(k): 1+2+3+...+k=k(k+1)/2
P(k+1): 1+2+3+...+k+(k+1)=(k+1)(k+2)/2
P(k+1): k(k+1)/2+(k+1)=(k+1)(k+2)/2
P(k+1): k(k+1)+2(k+1)/2=(k+1)(k+2)/2
P(k+1): (k+1)(k+2)/2=(k+1)(k+2)/2 (Adevarat) ==> P(k) adevarat ==> P(n) adevarat