Sửa đề: So sánh Q và 5/16
Ta có: \(Q=\frac15+\frac{2}{5^2}+\frac{3}{5^3}+\cdots+\frac{2022}{5^{2022}}\)
=>\(5Q=1+\frac25+\frac{3}{5^2}+\cdots+\frac{2022}{5^{2021}}\)
=>\(5Q-Q=1+\frac25+\frac{3}{5^2}+\cdots+\frac{2022}{5^{2021}}-\frac15-\frac{2}{5^2}-\cdots-\frac{2022}{5^{2022}}\)
=>\(4Q=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2021}}-\frac{2022}{5^{2022}}\)
Đặt \(A=\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2021}}\)
=>\(5A=1+\frac15+\cdots+\frac{1}{5^{2020}}\)
=>\(5A-A=1+\frac15+\cdots+\frac{1}{5^{2020}}-\frac15-\frac{1}{5^2}-\cdots-\frac{1}{5^{2021}}\)
=>\(4A=1-\frac{1}{5^{2021}}=\frac{5^{2021}-1}{5^{2021}}\)
=>\(A=\frac{5^{2021}-1}{4\cdot5^{2021}}\)
Ta có: \(4Q=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2021}}-\frac{2022}{5^{2022}}\)
\(=1+\frac{5^{2021}-1}{4\cdot5^{2021}}-\frac{2022}{5^{2022}}=1+\frac{5^{2022}-5-8088}{4\cdot5^{2022}}=1+\frac{5^{2022}-8093}{4\cdot5^{2022}}\)
=>\(4Q=1+\frac14-\frac{8093}{4\cdot5^{2022}}<\frac54\)
=>\(Q<\frac{5}{16}\)
