Question

so sánh (2+1)(2^2+1)(2^4+1)....(2^100+1)

và 2^100

Answer 1

gọi biểu thức \(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{100}+1\right)\) là A

Ta có:\(A=\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{100}+1\right)\)

\(\Rightarrow A=1+\dfrac{1}{2}+...+\dfrac{1}{2^{100}}\)

\(\Rightarrow2.A=2+1+\dfrac{1}{2}+...+\dfrac{1}{2^{99}}\)

\(\Rightarrow2A-A=\left(2+1+\dfrac{1}{2}+...+\dfrac{1}{2^{99}}\right)-\left(1+\dfrac{1}{2}+...+\dfrac{1}{2^{100}}\right)\)

\(\Rightarrow2-\dfrac{1}{2^{100}}< 2^{100}\)

hay \(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{100}+1\right)< 2^{100}\)