SOLUTION: The sixth & eleventh terms of a linear sequence are respectively 23 & 48. Calculate the sum of the first twenty terms
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Question 1134278: The sixth & eleventh terms of a linear sequence are respectively 23 & 48. Calculate the sum of the first twenty terms Found 2 solutions by ikleyn, greenestamps:Answer by ikleyn(52788) (Show Source):
There are 11-6 = 5 gaps between the 6-th and 11-th terms; so each gap (which is the common difference of the AP) is
= = 5.
Hence, the first term is
= = 23 - 5*5 = 23 - 25 = -2,
while the 20-th term is
= = -2 + 19*5 = 93.
The sum of the first 20 terms is 20 times the average of the 1-th and 20-th terms
= = = (-2+93)*10 = 91*10 = 910. ANSWERCHECK (Thanks to my MS Excel )
n
------------------
1 -2
2 3
3 8
4 13
5 18
6 23
7 28
8 33
9 38
10 43
11 48
12 53
13 58
14 63
15 68
16 73
17 78
18 83
19 88
20 93
Sum = 910
The 11th term is the 6th term plus the common difference 5 times, so the common difference is
The sum of the first 20 terms of the sequence is 20 times the average of all the terms; the average of all the terms is the average of the middle two terms -- the 10th and 11th terms. Since the 11th term is 48 and the common difference is 5, the 10th term is 43; then the average of the terms in the sequence is (43+48)/2 = 91/2.
