Question

Am nevoie urgent de ajutor daca se poate la o problema cu inductie:
"Demonstrati inegalitatea : n³ ≥ 3n²-2n, (∀) n≥1 si 2 la n >n²-n,(∀) n≥3"

Answer 1

  P(1)=>1^3>= 3*(1^2 )-2*1=> 1>=1,P(1) adevarata.
Presupunem ca P(k) :
Demonstram ca P(k) e adevarata:k ^3>=3k^2-2K.
formam P(k+1) :k+1 ;a a treia >=3*(k+1) la a doua-2(k+1)=>k^3+3k^2+3k+1>=3(k+1)^2-2(k+1)<=>k^3+3k^2+3k+1>=3k^2+6k+3-2k-1<=>k ^3+3k+1>=4k-1=>adevarat=>P(n) adevarata.
2 la n>n^2-n
P(3): 8>6 adevarat.
Presp.ca P(k) e adevarata.
P(k): 2 la k>k la a doua -k
Si formam P(k+1) 
2 la k+1>k+1 la a doua-(k+1)=> 2 la k*2>.k^2-2k+2-k-1 =>2(k+1)>k^2-2k+1-k-1 <=>2k+2>k^2-3k=>-k^2+5k+2>0,adevarata.