杭州交通信息网中央财经大学什么档次:有道数列题请教
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F(x)是R上增函数，且G(x)=F(x)-f(1-X)
1）设F(x)=X，若数列An满足a1=3,an=G(a(n-1)),试写出An的通项公式 题中n,n-1都试下标，表示第几项
1）设F(x)=X，若数列An满足a1=3,an=G(a(n-1)),试写出An的通项公式 题中n,n-1都试下标，表示第几项
G(x) = f(x) - f(1 - x)
f(x) = x
G(x) = x - (1 - x) = 2x - 1
a(n) = G(a(n-1)) = 2*a(n-1) - 1
a(1) = 3
a(n)
= 2*a(n-1) - 1
= 2^2*a(n-2) - (2^1 + 1)
= 2^3*a(n-3) - (2^2 + 2^1 + 1)
....
= 2^(n-1)*a(1) - [2^(n-2) + ....2^2 + 2^1 + 1]
= 3*2^(n-1) - [2^(n-1) - 1]
= 2^n + 1