设数列an满足:a1=1,且当n∈N*时,a(n)³+a(n)²(1-a(n+1))+1=a(n+1
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设数列an满足:a1=1,且当n∈N*时,a(n)³+a(n)²(1-a(n+1))+1=a(n+1)
比较a(n)与a(n+1)的大小
比较a(n)与a(n+1)的大小
![设数列an满足:a1=1,且当n∈N*时,a(n)³+a(n)²(1-a(n+1))+1=a(n+1](/uploads/image/z/18982211-59-1.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97an%E6%BB%A1%E8%B6%B3%EF%BC%9Aa1%3D1%2C%E4%B8%94%E5%BD%93n%E2%88%88N%2A%E6%97%B6%2Ca%EF%BC%88n%EF%BC%89%26%23179%3B%2Ba%EF%BC%88n%EF%BC%89%26%23178%3B%EF%BC%881-a%EF%BC%88n%2B1%EF%BC%89%EF%BC%89%2B1%3Da%EF%BC%88n%2B1)
a(n)³ + a(n)² * [1 - a(n + 1)] + 1= a(n + 1)
-> a(n)³ + a(n)² + 1= a(n + 1) * [1 + a(n)²]
-> a(n + 1) = [a(n)³ + a(n)² + 1] / [a(n)² + 1] = a(n)³ / [a(n)² + 1] + 1
又a1 = 1,所以 an ≥ 1
a(n + 1) / an = [a(n)³ + a(n)² + 1] / [a(n)³ + an]
an ≥ 1 -> a(n)² ≥ an -> a(n)² + 1 > an -> a(n)³ + a(n)² + 1 > a(n)³ + an
-> a(n + 1) / an > 1
-> a(n + 1) > an
-> a(n)³ + a(n)² + 1= a(n + 1) * [1 + a(n)²]
-> a(n + 1) = [a(n)³ + a(n)² + 1] / [a(n)² + 1] = a(n)³ / [a(n)² + 1] + 1
又a1 = 1,所以 an ≥ 1
a(n + 1) / an = [a(n)³ + a(n)² + 1] / [a(n)³ + an]
an ≥ 1 -> a(n)² ≥ an -> a(n)² + 1 > an -> a(n)³ + a(n)² + 1 > a(n)³ + an
-> a(n + 1) / an > 1
-> a(n + 1) > an
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