Question

so sánh hai số M và N biết:
M=2^2016 và N=(1+2)(2^2 + 1)(2^4 + 1)......(2^2008 + 1)

 

Answer 1

Ta có:

\(N=\left(1+2\right)\left(2-1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{2008}+1\right)\)

\(\Leftrightarrow N=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{2008}+1\right)\)

\(\Leftrightarrow N=\left(2^4-1\right)\left(2^4+1\right)...\left(2^{2008}+1\right)\)

\(\Leftrightarrow N=\left(2^8-1\right)...\left(2^{2008}+1\right)\)

\(\Leftrightarrow N=2^{4016}-1>2^{2016}=M\)

 

 

Answer 2

Ta có:

N=(1+2)(2−1)(22+1)(24+1)...(22008+1)N=(1+2)(2−1)(22+1)(24+1)...(22008+1)

⇔N=(22−1)(22+1)(24+1)...(22008+1)⇔N=(22−1)(22+1)(24+1)...(22008+1)

⇔N=(24−1)(24+1)...(22008+1)⇔N=(24−1)(24+1)...(22008+1)

⇔N=(28−1)...(22008+1)⇔N=(28−1)...(22008+1)

⇔N=24016−1>22016=M