始終搵唔到個通項,唔明白...

1,
3,4,6,9,13... = 3, 3+1, 3+1+2, 3+1+2+3, 3+1+2+3+4...
T(n)= 3+1+2+3+....+n-1
T(n)= 3+(1+n-1)(n-1)/2
T(n)= 3+(n)(n-1)/2

2,
1,3,7,13,21....= 1, 1+(2), 1+(2+4), 1+(2+4+6), 1+(2+4+6+8)...
1,3,7,13,21....= 1, 1+2(1), 1+2(1+2), 1+2(1+2+3), 1+2(1+2+3+4)...
T(n)=1+2(1+2+3+4+.....+(n-1))
T(n)=1+2(n)(n-1)/2
T(n)=1+(n)(n-1)

3,
3,6,12,24,48...=3, 3*2, 3*2^2, 3*2^3
等比數列, ratio=2, 首項是3
T(n)=3*[2^(n-1)]

4,
10,14,22,34,...=10, 10+4(1), 10+4(1+2), 10+4(1+2+3),......
T(n)=10+4(1+2+3+....+n-1)
T(n)=10+4n(n-1)/2
T(n)=10+2n(n-1)

5,
4,7,13,22,34,...=4, 4+3(1), 4+3(1+2), 4+3(1+2+3), 4+3(1+2+3+4)
T(n)=4+3(1+2+3+...+n-1)
T(n)=4+3n(n-1)/2
T(n)=4+(3/2)(n)(n-1)

其實只要注意項與項的相差...便會發現一些提示

1.18
2.31
3.96
4.50
5.49

(1)
T(2)-T(1)=4-3=1
T(3)-T(2)=6-4=2
T(4)-T(3)=9-6=3
T(5)-T(4)=13-9=4
Therefore T(n+1)-T(n)=n
T(n)=3+1+2+3+......+(n-1)=3+[1+(n-1)](n-1)/2=3+n(n-1)/2
Therefore the general term is T(n)=3+n(n-1)/2
Check:
T(1)=3+1(1-1)/2=3
T(2)=3+2(2-1)/2=3+1=4
T(3)=3+3(3-1)/2=3+3=6
T(4)=3+4(4-1)/2=3+6=9
T(5)=3+5(5-1)/2=3+10=13

(2)
T(2)-T(1)=3-1=2
T(3)-T(2)=7-3=4
T(4)-T(3)=13-7=6
T(5)-T(4)=21-13=8
Therefore T(n+1)-T(n)=2n
T(n)=1+2+4+6+8+......+2(n-1)=1+[1+(n-1)](n-1)=1+n(n-1)
Therefore the general term is T(n)=1+n(n-1)
Check:
T(1)=1+1(1-1)=1
T(2)=1+2(2-1)=1+2=3
T(3)=1+3(3-1)=1+6=7
T(4)=1+4(4-1)=1+12=13
T(5)=1+5(5-1)=1+20=21

(3)
T(2)/T(1)=6/3=2
T(3)/T(2)=12/6=2
T(4)/T(3)=24/12=2
T(5)/T(4)=48/24=2
It is a geometric sequence.
Therefore the general term is T(n)=3(2)^(n-1)

(4)
T(2)-T(1)=14-10=4
T(3)-T(2)=22-14=8
T(4)-T(3)=34-22=12
Therefore T(n+1)-T(n)=4n
Therefore T(n)=10+4+8+12+......+4(n-1)=10+2[1+(n-1)](n-1)=10+2n(n-1)
Therefore the general term is T(n)=10+2n(n-1)
Check:
T(1)=10+2(1)(1-1)=10
T(2)=10+2(2)(2-1)=10+4=14
T(3)=10+2(3)(3-1)=10+12=22
T(4)=10+2(4)(4-1)=10+24=34

You can try (5) yourselves.
The answer is T(n)=4+3n(n-1)/2

The formula of the sequence which is like:
a,a+b,a+b+(b+1),a+b+(b+1)+(b+2)...... is
T(n)=a+bn(n-1)/2

2006-11-07 16:54:46 補充：
我相信所有有明顯關係的數列都一定有通項連1,1,2,3,5,8,13,21,34......都有通項見http://zh.wikipedia.org/w/index.php?title=斐波那契数列&variant=zh-tw

2006-11-07 17:21:25 補充：
Sorry, some statement in my answer is wrong:The formula of the sequence which is like:a,a十b,a十b十2b,a十b十2b十3b...... isT(n)=a十bn(n-1)/2

2006-11-07 17:26:16 補充：
The sequence which is like:a,ab,ab(2b),ab(2b)(3b)......T(n)=(n-1)!ab^(n-1)The sequence which is like:ab,ab(b 1),ab(b 1)(b 2)......T(n)=a(b n-1)!/(b-1)!

一般ge等差/等比數列個d就明,不過好似以下個d數我點都搵唔到個通項...

1) 3,4,6,9,13,... 2) 1,3,7,13,21,... 3)3,6,12,24,48,...

4) 10,14,22,34,... 5) 4,7,13,22,34,...

唔該大家解釋一下點搵個通項,最好有埋個通項比我...

呢d數係唔係話找邏輯關係??
如果係：
1) 3,4,6,9,13,....就係+1,+2,+3,+4...之後應該係+5
(3+1=4,4+2=6,6+3=9,9+4=13,所以之後應該係13+5=18)

2) 1,3,7,13,21,...就係+2,+4,+6,+8 ...之後應該係+10
(1+2=3,3+4=7,7+6=13,13+8=21,所以之後應該係21+10=31)

3)3,6,12,24,48,...就係不斷乘2....之後都係乘2
(3乘2=6,6乘2=12,12乘2=24,24乘2=48,所以之後應該係48乘2=96)

4) 10,14,22,34....就係+4,+8,+12...之後應該係+16
(10+4=14,14+8=22,22+12=34,所以之後應該係34+16=50)

5) 4,7,13,22,34,...就係+3,+6,+9,+12...之後應該係+15
(4+3=7,7+6=13,13+9=22,22+12=34,所以之後應該係34+15=49)

通常你要睇下他們有冇關連,通常加減乘...有d會係次方...