Question

scrietu ca puterea a lui 2 numarul n+2,unde N=2+2²+2³+....+2²⁰¹⁴.​

Answer 1

Metoda I:

Observam ca avem o suma de termeni a unei progresii geometrice. Primul termen este 2, ratia este 2 iar numarul de termeni este 2014.

N = $N = 2\frac{2^{2014}-1}{2-1} = 2^{2015}-2\\N+2 = 2^{2015}-2+2=2^{2015}$

Metoda II:

Stim formula $1+ 2^1+2^2+2^3+...+2^n = 2^{n+1}-1$

Deci $N = 2^1+2^2+2^3+...+2^{2014} = 2^{2014+1}-2$

$N+2 = 2^1+2^2+2^3+...+2^{2014}+2 = 2^{2014+1}-2+2=2^{2015}$

Answer 2

$\it n= 2+2^2+2^3+\ ...\ +2^{2014}=2\cdot\dfrac{2^{2014}-1}{2-1}=2^{2015}-2\\ \\ \\ n+2=2^{2015}-2+2=2^{2015}$