中4 mathematical induction問題

2008-08-12 4:47 am
prove by mi that:
1+2+3+...+n=1/2n(n+1)
..............(之前有好多step,不過唔重要,我主要既問題係下面)
for any positive integerk,assume thatp(k) is true.ie> 1+2+3+...+k=1/2k(k+1)
then1+2+3+...+k+(k+1)=1/2k(k+1)+(k+1)
=1/2(k+1)(k+2) ←←←←


我就係唔明←←←←個度係點樣變出黎既.希望解答

同埋我都想問,1+2+3個d野係咪計數個時唔洗理佢?

回答 (3)

2008-08-12 5:16 am
✔ 最佳答案
1/2(k+1)(k+2) 呢個係因為 L.H.S 多左個k+1 ,,
就係為左証明點式係suitable for any positive integer,,,
因為L.H.S 多左個k+1,,,所以R.H.S 都要做一個1/2k(k+1) 裏面既k +1
而當prove的時候
L.H.S=1+2+3+...+k+k+1
=(1+2+3+...+k)+(k+1)
=1/2k(k+1)+(k+1)
=(k+1)x(1/2k+1)
=(k^2+k+2k+2)/2
=(k+1)(k+2)/2
就係咁...
至於你話1+2+3+...+k 係可以5駛計,,,因為你之前prove 左1+2+3+...+k=1/2k(k+1)
所以代翻落去就係
ps..其實主要係factorize 唸明 呢到k^2+k+2k+2 => (k+1)(k+2)就得
2008-08-12 5:23 am
因為1+2+3+...+k=1/2k(k+1)
(1+2+3+...+k)+(k+1)=1/2k(k+1) + (k+1)
之後要通分母,變成[k(k+1)+2(k+1)]/2
再抽(k+1) 就變成1/2(k+1)(k+2)

同埋我都想問,1+2+3個d野係咪計數個時唔洗理佢?
我唔明你問咩
不過做MI只要證明到P(1)同P(k+1) is true就得

ps.你既數底似乎麻麻地wo,做多d中2(抽common factor)同通分母既數對你有幫助,唔單止係MI呢個topic,日後其他topic都有幫助!加油吧!
2008-08-12 5:14 am

圖片參考:http://hk.geocities.com/stevieg_1023/0MIMI.gif

1+2+3+......係一句STATEMENT....可以整體咁去睇!


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