数列{An}中 A1=1,A2=2 An=An-1-An-2 (n ≥3) ,则A2008=?能求出通项公式吗
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![数列{An}中 A1=1,A2=2 An=An-1-An-2 (n ≥3) ,则A2008=?能求出通项公式吗](/uploads/image/z/12648052-28-2.jpg?t=%E6%95%B0%E5%88%97%7BAn%7D%E4%B8%AD+A1%3D1%2CA2%3D2+An%3DAn-1-An-2+%28n+%E2%89%A53%29+%2C%E5%88%99A2008%3D%3F%E8%83%BD%E6%B1%82%E5%87%BA%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%E5%90%97)
数列{An}中 A1=1,A2=2 An=An-1-An-2 (n ≥3) ,则A2008=?能求出通项公式吗
数列{An}中 A1=1,A2=2 An=An-1-An-2 (n ≥3) ,则A2008=?
能求出通项公式吗
数列{An}中 A1=1,A2=2 An=An-1-An-2 (n ≥3) ,则A2008=?能求出通项公式吗
A1=1 A7=1
A2=2 A8=2
A3=1 ……
A4=-1
A5=-2
A6=-1
A2008=A(6*334+4)=A4=-1
-1
解差分方程a(n+2) = a(n+1) - a(n), a(0) = -1, a(1) = 1
两边同时进行单边z变换
z^2 * ( A(z) - a(1)/z - a(0) ) = z ( A(z) - a(0) ) - A(z)
z^2 * A(z) - z + z^2 = zA(z) + z - A(z)
整理:
A(z) = ( 2z - z^2...
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解差分方程a(n+2) = a(n+1) - a(n), a(0) = -1, a(1) = 1
两边同时进行单边z变换
z^2 * ( A(z) - a(1)/z - a(0) ) = z ( A(z) - a(0) ) - A(z)
z^2 * A(z) - z + z^2 = zA(z) + z - A(z)
整理:
A(z) = ( 2z - z^2 ) / ( z^2 - z + 1 )
进行逆z变换
a(n) = ( -1 - i3^0.5 ) / 2 * ( 1 + i3^0.5 )^n / 2^n + ( -1 + i3^0.5 ) / 2 * ( 1 - i3^0.5 )^n / 2^n
= exp ( -i 2π/3 ) * exp ( i nπ/3 ) + exp ( i 2π/3 ) * exp ( -i nπ/3 )
= 2cos ( 2π/3 - nπ/3 )
即
a(n) = 2cos ( 2π/3 - nπ/3 )
a(2008) = 2cos ( 2π/3 - 2008π/3 )
= 2cos ( 2π/3 - 2008π/3 )
= 2cos ( 4π/3 ) = -1
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