Question

de demonstrat egalitatea;

1+6+20+...+ (2n-1) * 2 la puterea n-1 = 3+2 la puterea n * (2n - 3)

Answer 1

Demonstram egalitatea prin inductie matematica:
[tex]P(n):1+6+20+...+(2n-1)\cdot 2^{n-1}=3+2^{n}\cdot (2n-3)\\ P(1):1=3+2^{1}\cdot (2-3)(Adevarat)\\ Presupunem\ P(n)\ adevarat=>\ demonstram \ P(n+1)\ adevarat\\ P(n+1):1+6+20+...+(2n-1)\cdot 2^{n-1}+(2n+1)\cdot 2^n=\\ =3+2^{n+1}\cdot (2n-1)\\ 1+6+20+...+(2n-1)\cdot 2^{n-1}+(2n+1)\cdot 2^n=\\ =3+2^{n}\cdot (2n-3)+(2n+1)\cdot 2^n=3+2^n\cdot (4n-2)=\\ =3+2^{n+1}\cdot (2n-1) [/tex]