Question

Calculați 1+2 la puterea 1+2 la puterea 2+...+2 la puterea 2019.Vă rog să faceți desfășurat!!!

Answer 1

S = 1 + 2¹ + 2² + 2³ + ... + 2²⁰¹⁸ + 2²⁰¹⁹

S = 1 + 2¹ + 2² + 2³ + ... + 2²⁰¹⁸ + 2²⁰¹⁹ +

2²⁰²⁰ - 2²⁰²⁰

S = 1 + 2•(1+2¹+2²+2³+...+2²⁰¹⁸+2²⁰¹⁹) - 2²⁰²⁰

S = 1 + 2•S - 2²⁰²⁰

2•S - S = 2²⁰²⁰ - 1

=> S = 2²⁰²⁰ - 1

Answer 2

$1+2+2^2+...+2^{2019}=S\\ \\ \\ S+1=2+2+2^2+...+2^{2019}\\ \\ S+1=2^2+2^2+2^3+...+2^{2019}\\ \\ S+1=2^3+2^3+...+2^{2019}\\ \\ ...............................\\ \\ S+1=2^{2019}+2^{2019} \Rightarrow S=2^{2020}-1\\ \\ \boxed{2^a+2^a=2^a(1+1)=2^a\cdot 2^1=2^{a+1}}$

$\boxed{1+2+2^2+2^3+...+2^n=2^{n+1}-1}\\ \\ SAU:\\ \\ S=1+2+2+2^2+...+2^{2019} \ (*) \\ \\ 2S=2+2^2+2^3+...+2^{2020} \ (**)\\ \\ Scadem \ din \ relatia \ (**), \ relatia \ (*): \\ \\ 2S-S=(2+2^2+2^3+...+2^{2020})-(1+2+2+2^2+...+2^{2019})\\ \\ S=2^{2020}-1$