Question

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

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

và B = \(3^{128}-1\)

Answer 1

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

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

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

\(2A=\left(3^{16}-1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\left(3^{64}+1\right)\)

\(2A=\left(3^{32}-1\right)\left(3^{32}+1\right)\left(3^{64}+1\right)\)

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

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

Vậy \(A< B\)