點樣用數學歸納法證下列兩題 Fibonacci Sequence呢??高手求助!

2006-11-06 5:13 am

Demonstrate them clearly with examples and prove them algebraically for all n belonging to the positive integers.

Fibonacci Sequence

1). F (n+3)-F (n) = 2F (n+1)

2). F(n+2)+F(n-2)=3F(n)

回答 (2)

cheuk ting

2006-11-06 6:34 am

✔ 最佳答案

Fibonacci Sequence :
f(n+3) = f(n+1) + f(n+2) - - - - - (1)
f(n+2) = f(n) + f(n+1) - - - - - - - (2)
f(n+1) = f(n-1) + f(n) - - - - - - - (3)
f(n) = f(n -2) + f(n-1) - - - - - - - (4)

1) (1) + (2), we get :
f(n+3) + f(n+2) = f(n+1) + f(n+2) + f(n) + f(n+1)
f(n+3) = 2f(n+1) + f(n)
therefore, f(n+3) - f(n) = 2f(n+1)

2) (2) + (3) - (4), we get :
f(n+2) + f(n+1) - f(n) = f(n) + f(n+1) + f(n-1) + f(n) - f(n -2) - f(n-1)
f(n+2) - f(n) = 2f(n) - f(n-2)
therefore, f(n+2) + f(n-2) = 3f(n)

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K. L.

2006-11-10 6:51 am

但沒有用數學歸納法

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