Question

So sánh hai số bằng cách vận dụng hằng đẳng thức : 

\(A=4.\left(3^2+1\right).\left(3^4+1\right)....\left(3^{64}+1\right)\) và 

\(B=3^{128}-1\)

Answer 1

\(A=4\left(3^2+1\right)\left(3^4+1\right)....\left(3^{64}+1\right)\)

\(2A=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)....\left(3^{64}+1\right)\)

\(2A=\left(3^4-1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)

\(2A=\left(3^{16}-1\right)\left(3^{16}+1\right)\left(3^{22}+1\right)\left(3^{64}+1\right)\)

\(2A=\left(3^{64}-1\right)\left(3^{64}+1\right)\)

\(2A=3^{128}-1\Rightarrow A=\frac{3^{128}-1}{2}< 3^{128}-1=B\)

Vậy \(A< B\)

Chúc bạn học tốt !!!

Answer 2

A.(3²-1)=4.(3²-1)(3²+1)(3⁴+1)...(3⁶⁴+1)=4.(3⁴-1)(3⁴+1)...(3⁶⁴+1)=  ...  =4.(3¹²⁸-1)

<=>8A=4B <=>2A=B =>B>A