Question

1+2+2^2+2^3+. +2^n=2^2018-1.

Answer 1

Explicație pas cu pas:

Metoda 1:

$S=1+2+2^2+...+2^n\\2S=2+2^2+2^3+...+2^{n+1}\\2S-S=2^{n+1}+...+2^3+2^2+2-(2^n+...+2^2+2+1)=2^{n+1}-1\\\boxed{S=2^{n+1}-1}\\\\2^{n+1}-1=2^{2018}-1\\2^{n+1}=2^{2018}\\n+1=2018\\n=2017$

Metoda 2:

$\boxed{S_n=b_1+b_2+...+b_n=b_1 \cdot \frac{q^{n}-1}{q-1} }$

$b_1=2^0=1\\q=2\\\underbrace{1+2+2^2+...+2^n}_{n+1 \ termeni}$

$1+2+2^2+...+2^n=1 \cdot \frac{2^{n+1}-1}{2-1} =2^{n+1}-1$

$2^{n+1}-1=2^{2018}-1\\2^{n+1}=2^{2018} \implies n=2018-1\\n=2017$