SOLUTION: What is the common difference? An arithmetic series has 9 terms with a sum of -135. It has the first term of 9.

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Question 1162999: What is the common difference? An arithmetic series has 9 terms with a sum of -135. It has the first term of 9.
Found 3 solutions by ikleyn, MathTherapy, greenestamps:
Answer by ikleyn(52788) About Me  (Show Source):
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.

The sum of an arithmetic progression is the average of its first and last terms times the number of terms.


Hence, the average of the first and the last terms is a quotient of division its sum by the number of terms.


In our case, the average of the first and the last terms is  -135%2F9 = -15:
 9+39 =

    %28a%5B1%5D%2Ba%5B9%5D%29%2F2 = -15,  or  %289%2Ba%5B9%5D%29%2F2 = -15.


It implies


    a%5B9%5D = (-15)*2 - 9 = -30 - 9 = -39.    


The distance on the number line between the first term 9 and the 9-th term -39 is 9 + 39 = 48.


There are 8 equal intervals (gaps) between the first and the last terms, so the value of each gap is  48%2F8 = 6.


Hence (and since the sequence is decreasing), the common difference of the AP is -6.    ANSWER

Solved.

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

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

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


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.




Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
What is the common difference? An arithmetic series has 9 terms with a sum of -135. It has the first term of 9. 
Sum-of-an-A.P. formula: matrix%281%2C3%2C+S%5Bn%5D%2C+%22=%22%2C+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29%29

matrix%281%2C3%2C+S%5B9%5D%2C+%22=%22%2C+%289%2F2%29%282%289%29+%2B+%289+-+1%29d%29%29 -- Substituting matrix%282%2C3%2C+9%2C+for%2C+n%2C+9%2C+for%2C+a%5B1%5D%29 

matrix%281%2C3%2C++-+135%2C+%22=%22%2C+%289%2F2%29%2818+%2B+8d%29%29 ------- Substituting matrix%281%2C3%2C+-+135%2C+for%2C+S%5B9%5D%29

- 270 = 9(18 + 8d) ------- Cross-multiplying

matrix%281%2C3%2C+%28-+270%29%2F9%2C+%22=%22%2C+18+%2B+8d%29

- 30 = 18 + 8d ====> - 48 = 8d 

Common difference, or highlight_green%28matrix%281%2C5%2C+d%2C+%22=%22%2C+%28-+48%29%2F8%2C+%22=%22%2C+-+6%29%29

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


The sum of the terms in an arithmetic sequence is the number of terms, multiplied by the average of the terms. So the average of all the terms is the sum, divided by the number of terms.

In this problem, the number of terms is 9 and the sum is -135, so the average of the terms is -135/9 = -15.

In an arithmetic sequence of 9 terms, the average of all the terms is the term in the middle -- the 5th term.

So the 5th term in the sequence is -15.

The 5th term is the first term, plus the common difference 4 times:

-15+=+9%2B4d
4d+=+-24
d+=+6

ANSWER: The common difference is -6