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): You can put this solution on YOUR website!
.
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.     ANSWER


CHECK   (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

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    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions
    - Mathematical induction and arithmetic progressions
    - Mathematical induction for sequences other than arithmetic or geometric

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Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


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.

And so the sum of the first 20 terms is
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