Question 1162999
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<pre>

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  {{{-135/9}}} = -15:
 9+39 =

    {{{(a[1]+a[9])/2}}} = -15,  or  {{{(9+a[9])/2}}} = -15.


It implies


    {{{a[9]}}} = (-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  {{{48/8}}} = 6.


Hence (and since the sequence is decreasing), the common difference of the AP is -6.    <U>ANSWER</U>
</pre>

Solved.


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For introductory lessons on arithmetic progressions see 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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



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.