运用乘法公式计算(3+1)(3^2+1)(3^4+1)(3^8+1)(3^16+1)
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设S=原式=(3+1)(3^2+1)(3^4+1)(3^8+1)(3^16+1)
两边同乘以(3-1)得:
(3-1)S
=(3-1)(3+1)(3^2+1)(3^4+1)(3^8+1)(3^16+1)
=(3^2-1)(3^2+1)(3^4+1)(3^8+1)(3^16+1)
=(3^4-1)(3^4+1)(3^8+1)(3^16+1)
=(3^8-1)(3^8+1)(3^16+1)
=(3^16-1)(3^16+1)
=(3^32-1)
所以
2S=(3^32-1)
原式=(3^32-1)/2
两边同乘以(3-1)得:
(3-1)S
=(3-1)(3+1)(3^2+1)(3^4+1)(3^8+1)(3^16+1)
=(3^2-1)(3^2+1)(3^4+1)(3^8+1)(3^16+1)
=(3^4-1)(3^4+1)(3^8+1)(3^16+1)
=(3^8-1)(3^8+1)(3^16+1)
=(3^16-1)(3^16+1)
=(3^32-1)
所以
2S=(3^32-1)
原式=(3^32-1)/2
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