SOLUTION: The six term of an arithmetic progression is 23 and the sum of the first six terms is 78.Find
(a)the common difference and the first term
(b)the tenth term
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-> SOLUTION: The six term of an arithmetic progression is 23 and the sum of the first six terms is 78.Find
(a)the common difference and the first term
(b)the tenth term
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Question 1094753: The six term of an arithmetic progression is 23 and the sum of the first six terms is 78.Find
(a)the common difference and the first term
(b)the tenth term Found 2 solutions by ikleyn, greenestamps:Answer by ikleyn(52788) (Show Source):
You can put this solution on YOUR website! .
The sixth term of an arithmetic progression is 23 and the sum of the first six terms is 78.Find
(a)the common difference and the first term
(b)the tenth term
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Since the sum of the six terms is 78 and the 6-th term is 23, the sum of the first 5 terms is 78-23 = 55.
In turn, it means that the 3-rd term, which is exactly in the middle of that five terms, is = 11.
Thus we know that = 11 and = 23. The difference - = 23-11 = 12 is exactly 3 times the common difference.
Hence, the common difference is = 4.
Now you know ALL about this progression and can answer any question on your own.
You can put this solution on YOUR website! A key concept in working with arithmetic sequences and series is that you can always group the terms in pairs so that the sum in each pair is the same; or, if the number of terms in the sequence is odd, there will be a single term in the middle that is half of that common sum.
So here is how I would work this problem....
The sum of the first 6 terms is 78. That means there are 3 pairs of terms, with each pair having a sum of 78/3 = 26.
The 6th term is 23; it pairs up with the first term; and the sum of the first and 6th terms is 26. That means the first term is 26-23 = 3.
The 6th term, 23, is the first term, 3, plus the common difference 5 times:
We are done with part (a): the common difference is 4; the first term is 3.
Part (b) is now easy. The 10th term is the first term, plus the common difference 9 times:
