Question

Answer 1

a) \(A=1+2^1+2^2+...+2^{2019}\)

\(\Rightarrow2A=2+2^2+2^3+...+2^{2020}\)

\(\Rightarrow A=2A-A=2+2^2+2^3+...+2^{2020}-1-2-2^2-...-2^{2019}=2^{2020}-1\)

\(\Rightarrow A+1=2^{2020}-1+1=2^{2020}=\left(2^{1010}\right)^2\) là số chính phương

b) \(B=1+3+3^3+...+3^{2020}+3^{2021}\)

\(\Rightarrow3B=3+3^2+3^3+...+3^{2022}\)

\(\Rightarrow2B=3B-B=3+3^2+3^3+...+3^{2022}-1-3-3^2-....-3^{2021}=3^{2022}-1\)

\(\Rightarrow2B+1=3^{2022}-1+1=3^{2022}=\left(3^{1011}\right)^2\) là số chính phương