Question

 Rút gọn biểu thức:

A = (3 + 1) (3² + 1) (3⁴ + 1) ... (3⁶⁴ + 1)

 

Answer 1

A = (3 + 1) (3² + 1) (3⁴ + 1) ... (3⁶⁴ + 1)

2A = (3 - 1)(3 + 1) (3² + 1) (3⁴ + 1) ... (3⁶⁴ + 1)

2A = (3² - 1)(3² + 1) (3⁴ + 1) ... (3⁶⁴ + 1)

= (3⁴ - 1)(3⁴ + 1) ... (3⁶⁴ + 1)

= (3⁸ - 1)(3⁸ + 1)(3¹⁶+1)(3³²+1)(3⁶⁴+1)

= (3¹⁶-1)(3¹⁶+1)(3³²+1)(3⁶⁴+1)

= (3³²-1)(3³²+1)(3⁶⁴+1)

= (3⁶⁴-1)(3⁶⁴+1)

= (3¹²⁸-1)

=> A = \(\frac{3^{128}-1}{2}\)

Answer 2

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

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

        áp dụng hằng đẳng thức \(a^2-b^2\)

ta có 2A=\(3^{128}-1\)=>A=\(\frac{3^{128}-1}{2}\)