Question

sa se determine n aparține N astfel incat: a) 1+2+4+8+...+2*2^n=131071

Answer 1

[tex]1+2+4+8+_{\dots}+2\cdot 2^{n}=131071\\ 1+2+2^2+2^3+_{\dots}+2^{n+1}=131071\\ \text{Dupa cum se poate observa,partea stanga este o progresie geometrica}\\ \text{cu}\ b_1=\text{1 si q=2}:\\ S=b_1\cdot \dfrac{q^{n}-1}{q-1}\\ \dfrac{2^{n+2}-1}{2-1}=131071\\ 2^{n+2}-1=131071\\ 2^{n+2}=131072\\ 2^{n}\cdot 4=131072\\ 2^{n}=32768\\ 2^{n}=2^{15}\Rightarrow \boxed{n=15}[/tex]