推 attomahawk :推薦這個解法！ 03/28 19:21
※ 引述《barein (好禮三選一)》之銘言：
: 今天在報紙上看到的
: A1=0 A2=12 .....An= (An-1+An-2)/2 即 A3=6 A4=9
: 試問 n→無限大時 An為多少？
A(n) = (A(n-1)+A(n-2)/2
A(n) - A(n-1) = -(A(n-1)-A(n-2))/2 = (A(n-2)-A(n-3))/4 = ...
= (A(2)-A(1)) (-1/2)^(n-2) = 12 (-1/2)^(n-2) ---> (1)
A(n) + 0.5A(n-1) = A(n-1) + 0.5A(n-2)) = A(n-2) + 0.5A(n-3) = ...
= A(2) + 0.5A(1) = 12 ---> (2)
(1) + 2*(2) = 3A(n) = 12 (-1/2)^(n-2) + 24
A(n) = 4 (-1/2)^(n-2) + 8
n→無限大時, A(n) = 8.
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