运用乘法公式计算(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)的值
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![运用乘法公式计算(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)的值](/uploads/image/z/474618-66-8.jpg?t=%E8%BF%90%E7%94%A8%E4%B9%98%E6%B3%95%E5%85%AC%E5%BC%8F%E8%AE%A1%E7%AE%97%EF%BC%882%2B1%EF%BC%89%EF%BC%882%26%23178%3B%2B1%EF%BC%89%EF%BC%882%5E4%2B1%EF%BC%89%C2%B7%EF%BC%882%5E8%2B1%EF%BC%89...%EF%BC%882%5E2n%2B1%EF%BC%89%E7%9A%84%E5%80%BC)
运用乘法公式计算(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)的值
运用乘法公式计算(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)的值
运用乘法公式计算(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)的值
(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)
=(2-1)(2+1)(2²+1)(2^4+1)·(2^8+1)...(2^2n+1)
=2^4n-1
(a-b)(a+b)=a^2-b^2,1^2=1, 2-1=1
将该式乘以(2-1)=1则其结果不变,但是可依次化简:
(2-1)*(2+1)=(2^2-1),(2^2-1)(2^2+1)=(2^4-1)..........
原式=2^4n-1