標題:
Arithmetic sequence
發問:
1. The sum of the 3rd term and the 4th term of an arithmetic sequence is -1 and the sum of the first 15 terms is 195. Find the sum of the first 26 terms of the sequence.2. Given that the sum of the first 10 terms of an arithmetic sequence is 220. and the sum of the first 15 terms is 180. Find a) the first term... 顯示更多 1. The sum of the 3rd term and the 4th term of an arithmetic sequence is -1 and the sum of the first 15 terms is 195. Find the sum of the first 26 terms of the sequence. 2. Given that the sum of the first 10 terms of an arithmetic sequence is 220. and the sum of the first 15 terms is 180. Find a) the first term and the common difference b) the value of k, if the sum of the first k terms of the sequence is 216.
最佳解答:
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： 可參考以下網址
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