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\(\frac{1.2.4+2.4.8+4.8.16+...+128.256.512}{1.3.4+2.6.8+4.12.16+...+128.384.512}\)
\(=\frac{1.2.4.\left(1^3+2^3+4^3+8^3...+128^3\right)}{1.3.4.\left(1^3+2^3+4^3+8^3...+128^3\right)}\)
\(=\frac{2}{3}\)
Vì 2/3 = 8/12 < 9/12 = 3/4
\(\Rightarrow\frac{1.2.4+2.4.8+4.8.16+...+128.256.512}{1.3.4+2.6.8+4.12.16+...+128.384.512}< \frac{3}{4}\)
\(M=\frac{1.2.4+2.4.8+4.8.16+8.16.32}{1.3.4+2.6.8+4.12.16+8.24.32}\)
\(M=\frac{\left(1.2.4\right).1^3+\left(1.2.4\right).2^3+\left(1.2.4\right).4^3+\left(1.2.4\right).8^3}{\left(1.3.4\right).1^3+\left(1.3.4\right).2^3+\left(1.3.4\right).4^3+\left(1.3.4\right).8^3}\)
\(M=\frac{\left(1.2.4\right).\left(1^3+2^3+4^3+8^3\right)}{\left(1.3.4\right).\left(1^3+2^3+4^3+8^3\right)}\)
M = \(\frac{2}{3}\)
\(\frac{2.4+2.4.8+4.8.16+8.16.32}{3.4+2.6.8+4.12.16+8.24.32}=\frac{2.4+2.4.2.2.2.2+4.2.4.4.4+4.2.2.4.2.4.8}{3.4+2.3.2.4.2+4.3.4.4.4+4.2.4.3.2.4.8}\)
\(=\frac{2.4\left(1+2.2.2+4.4.4+2.4.2.4.8\right)}{3.4\left(1+2.2.2+4.4.4+2.4.2.4.8\right)}\)
\(=\frac{2.4}{3.4}\)
\(=\frac{2}{3}\)