General Term ! ! !

Let the name of the sequence be sequence A
Let T(1) = 3 , T(2) = 9 , T(3) = 11 , T(4) = 17 , T(5) = 19 , T(6) = 25

T(2) - T(1) = 9 - 3 = 6
T(3) - T(2) = 11 - 9 = 2
T(4) - T(3) = 17 - 11 = 6
T(5) - T(4) = 19 - 17 = 2
T(6) - T(5) = 25 - 19 = 6

咁樣好難直接睇得出佢嘅general term。

不如咁樣，我哋首先將sequence A拆開做sequence B和sequence C，其中sequence B取sequence A的第單數項，其中sequence C取sequence A的第雙數項，如下：
sequence B: 3, 11, 19, …
sequence C: 9, 17, 25, …


不難看出，sequence B和sequence C都是arithmetic sequence，而且佢哋嘅common difference都是8。

但咁樣都仲未夠，我哋仲要喺sequence B和sequence C嘅項與項之間各加上一項，形成sequence D和sequence E，使sequence A的第單數項是等價於取自sequence D的第單數項，使sequence A的第雙數項是等價於取自sequence E的第雙數項。

其實加上甚麼項都可以，因為我最後會有啲方法整到佢哋無效。不過既然sequence B和sequence C都是arithmetic sequence，而arithmetic sequence嘅general term確實咁容易搵，咁我哋要整到sequence D和sequence E仍然是arithmetic sequence。

整完出嚟嘅sequence D和sequence E係咁樣嘅：

sequence D: 3, 7, 11, 15, 19, 23, …
sequence E: 5, 9, 13, 17, 21, 25, …

the general term T1(n) of sequence D
= 3 + (n - 1)4
= 3 + 4n - 4
= 4n - 1

the general term T2(n) of sequence E
= 5 + (n - 1)4
= 5 + 4n - 4
= 4n + 1

∴the general term T(n) of sequence A

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/T_1n_T_2n.jpg


但這表達式肯定不能令人滿足，所以我哋要進一步簡化至唔駛分case。

現在的要求是：當n是單數，就用T1(n)的值；當n是雙數，就用T2(n)的值。

但是合併case，就要兩個都要同時用，咁點算呢？

我哋可以咁樣解決嘅：無論n是單數還是雙數，T1(n)和T2(n)的值都同時用哂。只不過當n是單數的時候，我哋就特登做啲手腳，令到T2(n)的值用咗等於冇用；當n是雙數的時候，我哋就特登做啲手腳，令到T1(n)的值用咗等於冇用。

點樣用咗等於冇用呀？

最直接的方法當然是把T(n)表達至這樣式：
T(n) = T1(n)h1(n) + T2(n)h2(n)

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/h_1n_h_2n.jpg


當然這樣不能直接寫做答案啦！我哋仲要搵埋T1(n)同埋T2(n)嫁！


嚟到呢度，令我諗起(- 1)ⁿ。

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/-1n.jpg


這正正符合T1(n)和T2(n)的要求。

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/h_1n_h_2n_answer.jpg


Hence the general term T(n) of sequence A

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/final.jpg



∴the general term of the sequence 3 , 9 , 11 , 17 , 19 , 25 , ... is T(n) = 4n + (- 1)ⁿ

4n+(-1)的n次方

2007-12-15 17:51:02 補充：
T(1) 1,T(2)-1,T(3) 1,T(4)-1,...=4,8,12,16,...So, the general term is 4n (-1)的n次方

1st term: 3
2nd term: 9
9-3=6

2nd term: 9
3rd term: 11
11-9=2

3rd term: 11
4th term: 17
17-11=6

4th term: 17
5th term: 19
19-17=2

6, 2, 6, 2, 6, 2, 6, ...
3, 9, 11, 17, 19, 25, 27, 33, 35, ...

If the current term number is an odd number, then the next number is the current number plus two.
If the current term number is an even number, then the next number is the current number plus six.

I don't know the equation of this squence, but I think this rule can help you.