Question

Să se verifice prin inducție:
2ⁿ > 2n + 3, n >= 4​

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

$\it P(n):\ \ 2^n>2n+1,\ \ \forall\ n\geq4\\ \\ I)\ P(4):\ \ 2^4=2\cdot4+3 \Leftrightarrow 16>11\ (A)\\ \\ II)\ \ P(k)\Rightarrow P(k+1)\\ \\ P(k): 2^k>2k+3\\ \\ P(k+1):\ \ 2^{k+1}>2(k+1)+3\\ \\ P(k):\ \ 2^k>2k+3|_{\cdot2}\Rightarrow 2^{k+1}>4k+6>2k+5 \Rightarrow\\ \\ \Rightarrow 2^{k+1}>2k+2+3 \Rightarrow 2^{k+1}>2(k+1)+3 \Rightarrow P(k+1)\\ \\Prin\ urmare,\ \ P(k)\Rightarrow P(k+1),\ \forall\ k>4,\ deci\ P(n)\ adev\breve arat\breve a$