Question 1140687
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<pre>
The last five terms of this progression are  {{{a[11]}}},  {{{a[12]}}},  {{{a[13]}}},   {{{a[14]}}}  and  {{{a[15]}}}.


The middle term of these 5 terms is  {{{a[13]}}}, and it is equal to   {{{1/5}}}  of the sum of that five terms.


     It is the key point in the solution, and you should clearly understand it:

          the average of any odd number of sequential terms of an AP is  the middle term of this partial sequence.



Thus  {{{a[13]}}} = {{{305/5}}} = 61.


Now we know  {{{a[13]}}} = 61  and  {{{a[6]}}} = 26.


Between  {{{a[13]}}} and  {{{a[6]}}},  there are 13-6 = 7 gaps, each of which is equal to the common difference of the AP.


Hence, the common difference  d = {{{(61-26)/7}}} = {{{35/7}}} = 5.


Then the first term is  {{{a[1]}}} = {{{a[6]}}} - 5*5 = 26 - 25 = 1.


     the 15-th term is  {{{a[15]}}} = {{{a[1]}}} + (15-1)*d = 1 + 14*5 = 71.


Now the sum of the 15 terms of the AP is  {{{((a[1]+a[15])/2)*15}}} = {{{((1+71)/2)*15}}} = 540.
</pre>

Solved.


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There is a bunch of lessons on arithmetic progressions in this site:

&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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A>

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Calculating-partial-sums-of-arithmetic-progressions.lesson>Calculating partial sums of arithmetic progressions</A> 

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A>


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.