Arithmetic sequence

1. the 3rd term= a + (3-1)d =a+2d
the 4th term =a +(4-1)d =a+3d
the sum of 3rd term and the 4th term =a+2d+a+3d=-1
2a+5d=-1..........(1)

To sum up the terms of an arithmetic sequence

圖片參考：http://www.mathsisfun.com/algebra/images/partial-sum-h.gif
now n=15

15/2 (2a+14d) =195
15a+105d=195...........(2)

(1) x 21 -(2)
42a +105d -15a-105d=-21-195
27a=-216
a=-8
sub 1=-8 into (1)
2(-8)+5d=-1
-16+1=-5d
-15=-5d
d=3
To sum up the terms of an arithmetic sequence

圖片參考：http://www.mathsisfun.com/algebra/images/partial-sum-h.gif
n=26
=26/2[2(-8)+25(3)]
=13(-16+75)
=767 //

2. n=10
10/2 (2a+9d) =220
20a+90d=440
2a+9d=44......(1)

n=15
15/2(2a+14d)=180
15a+105d=180
3a+21d=36
a+7d=12........(2)

(1)-2x(2)
2a+9d-2a-14d=44-24
-5d=20
d=-4
sub d=-4 into (1)
2a+9(-4)=44
a-18=22
a=40
2a) the first term a=40, and the common difference d=-4
2b)Let n=k
To sum up the terms of an arithmetic sequence

圖片參考：http://www.mathsisfun.com/algebra/images/partial-sum-h.gif
k/2=[2(40)+(k-1)(-4)]
k/2=(80-4k+4)
k=2(84-4k)
k=168-8k
5k=160
k=32 //




2011-08-22 18:45:41 補充：
Sorry, 2(b) is wrong, correction:
Let n=k,
k/2[2(40)+(k-1)(-4)]=216
k/2(80-4k+4)=216
84k-4k^2=432
k^2-21k+108=0
(k-12)(k-9)=0
k=12// or k=9//

2011-08-22 18:50:40 補充：
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