Bài 1:
\(A=3+3^2+3^3+...+3^{2017}+3^{2018}\)
=>\(3\cdot A=3^2+3^3+3^4+...+3^{2018}+3^{2019}\)
=>\(3\cdot A-A=3^2+3^3+...+3^{2018}+3^{2019}-3-3^2-3^3-...-3^{2017}-3^{2018}\)
=>\(2A=3^{2019}-3\)
=>\(2A+3=3^{2019}\)
=>\(3^x=3^{2019}\)
=>x=2019
Bài 2:
\(A=3+3^2+...+3^{99}+3^{100}\)
=>\(3\cdot A=3^2+3^3+...+3^{100}+3^{101}\)
=>\(3\cdot A-A=3^2+3^3+...+3^{100}+3^{101}-3-3^2-...-3^{99}-3^{100}\)
=>\(2A=3^{101}-3\)
=>\(2A+3=3^{101}\)
mà \(2A+3=3^n\)
nên \(3^n=3^{101}\)
=>n=101
Bài 3:
\(A=5+5^2+5^3+...+5^{99}+5^{100}\)
=>\(5A=5^2+5^3+5^4+...+5^{100}+5^{101}\)
=>\(5A-A=5^2+5^3+...+5^{100}+5^{101}-5-5^2-5^3-...-5^{99}-5^{100}\)
=>\(4\cdot A=5^{101}-5\)
=>\(4A+5=5^{101}\)
mà \(4A+5=5^x\)
nên \(5^x=5^{101}\)
=>x=101
Bài 4:
\(A=6^2+6^3+6^4+...+6^{2019}+6^{2020}\)
=>\(6A=6^3+6^4+6^5+...+6^{2020}+6^{2021}\)
=>\(6A-A=6^3+6^4+...+6^{2020}+6^{2021}-6^2-6^3-...-6^{2019}-6^{2020}\)
=>\(5A=6^{2021}-6^2=6^{2021}-36\)
=>\(6^{43x}-36=6^{2021}-36\)
=>43x=2021
=>x=2021/43=47
