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

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
~~~~~~~~~~~~~~~~~~~~~~

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.

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On arithmetic progressions, see the introductory lessons
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

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    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".

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Free of charge online textbook in ALGEBRA-II
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Answer by greenestamps(13200)   (Show Source): 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:

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