\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)....\left(\frac{1}{98^2}-1\right)\left(\frac{1}{99^2}-1\right)\)
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S
2 tháng 12 2019
Đặt \(A=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^3+...+\left(\frac{1}{2}\right)^{99}\)
\(\Rightarrow A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}\)
\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{98}}\)
\(\Rightarrow2A-A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{98}}-\frac{1}{2}-\frac{1}{2^2}-\frac{1}{2^3}-...-\frac{1}{2^{99}}\)
\(\Rightarrow A=1-\frac{1}{2^{99}}=\frac{2^{99}-1}{2^{99}}\)
28 tháng 12 2019
\(=\frac{1-2^2}{2^2}.\frac{1-3^2}{3^2}.\frac{1-4^2}{4^2}...\frac{1-98^2}{98^2}.\frac{1-99^2}{99^2}\)
\(=\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}...\frac{98^2-1}{98^2}.\frac{99^2-1}{99^2}\)
= \(\frac{\left(2-1\right).\left(2+1\right)}{2^2}.\frac{\left(3-1\right).\left(3+1\right)}{3^2}.\frac{\left(4-1\right).\left(4+1\right)}{4^2}...\frac{\left(98-1\right)\left(98+1\right)}{98^2}.\frac{\left(99-1\right)\left(99+1\right)}{99^2}\)
\(=\frac{\left(2-1\right).\left(3-1\right).\left(4-1\right)...\left(99-1\right)}{2.3.4...98.99}.\frac{\left(2+1\right).\left(3+1\right).\left(4+1\right)...\left(99+1\right)}{2.3.4...98.99}\)
\(=\frac{1.2.3....98}{2.3.4...98.99}.\frac{3.4.5...100}{2.3.4...98.99}\)
\(=\frac{1}{99}.\frac{100}{2}\)
\(=\frac{50}{99}\)
Chúc bạn học tốt !!!
A=50/99