【易】中四a maths

2007-09-07 2:25 am
利用數學歸納法證明,其中n是自然數:
1x2+2x3+3x4+ ... +n(n+1)=1除3 n(n+1)(n+2)
thx

回答 (3)

2007-09-07 2:32 am
✔ 最佳答案
設P(n)為命題, "1x2+2x3+3x4+ ... +n(n+1)=(1/3) n(n+1)(n+2)"
當n=1
左方=1x(1+1)=2
右方=(1/3)(1)(1+1)(1+2)=6/3=2=左方
所以P(1)成立
設對於正整k, P(k) 成立 , 即"1x2+2x3+...+k(k+1)=(1/3)(k)(k+1)(k+2)"
當n=k+1 ,
左方
=1x2+2x3+...+k(k+1)+(k+1)(k+2)
=(1/3)(k)(k+1)(k+2)+(k+1)(k+2)
=(1/3)(k)(k+1)(k+2)+(1/3)(3)(k+1)(k+2)
=(1/3)(k+1)(k+2)(k+3)
=右方
所以P(k+1)成立.
根據數學歸納法, 對於正整數n , P(n) 成立
2007-09-07 2:50 am
設P(n)為命題"1x2+2x3+3x4+ ... +n(n+1)=1除3 n(n+1)(n+2)"
當n=1時,
左方:1(1+1)=2
右方:(1/3)(1)(1+1)(1+2)=2
由於左方=右方
所以p(1)成立
設命題p(k)成立,即:"1x2+2x3+3x4+ ... +k(k+1)=1除3 k(k+1)(k+2)"
當n=(k+1)時
左方:(1/3)(k)(k+1)(k+2)+(k+1)(k+2)
=(k+1)(k+2)(1/3k+1)
=(k+1)(k+2) [(k+3) / 3]
=(1/3)(k+1)(k+2)(k+3)
右方:(1/3) (k+1)(k+1+1)(k+1+2)
=(1/3)(k+1)(k+2)(k+3)
由於左方=右方
所以命題p(k+1)成位
根據數學歸納法,命題p(n)對所有自然數都成位。
參考: 如錯讀提點
2007-09-07 2:35 am
設S(n)為命題
"1x2+2x3+3x4+ ... +n(n+1)=1/3 n(n+1)(n+2)"
當n=1,
LHS=1x2=2
RHS=1/3(1)(2)(3)=6
S(1)成立
假設對於自然數k,S(k)成立
即1x2+2x3+3x4+...+k(k+1)=1/3 k(k+1)(k+2)
當n=k+1
LHS=1x2+2x3+3x4+...+k(k+1)+(k+1)(k+1+1)
=1/3 k(k+1)(k+2)+(k+1)(k+2)
=(k+1)(k+2)(1/3k +1)
=1/3(k+1)(k+2)(k+3)
=RHS
根據數學歸納法的原理,對於所有自然數n,S(n)成立

2007-09-06 18:37:04 補充:
當n=1RHS=1/3(1)(2)(3)=2打錯野-.-


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