Question 1134773
.
<pre>

Notice that from the condition, the sum of the first NINE terms is 36 + 90 = 126.


Now from the condition we have


      a)  the sum of the first 5 terms is  36,     (1) 
and
      b)  the sum of the first 9 terms is 126.     (2)


For any AP,  the sum of its first n terms is n times the arithmetic mean of the first and the n-th terms - 

therefore, for the given AP the arithmetic mean of the first and the 9-th terms is 126/9 =  14.


But in the given case this arithmetic mean of the first and the 9-th terms is nothing else as the 5-th term.


    Thus the 5-th term of the given AP is  14.



Similarly,  for the given AP the arithmetic mean of the first and the 5-th terms is 36/5 =  7.2.

But in this case this arithmetic mean of the first and the 5-th terms is nothing else as the 3-rd term.


    Thus the 3-rd term of the given AP is  7.2.


Between the 3-rd and the 5-th terms there are two equal gaps, each equal to the common difference.


So, the common difference of the given AP is  {{{(14-7.2)/2}}} = {{{6.8/2}}} = 3.4.



Now we know that the 3-rd term of the AP is 7.2 and the common difference is 3.4.

Hence, its 1-st term is  7.2 - 2*3.4 = 7.2 - 6.8 = 0.4.


Now we know ALL about this AP:  {{{a[1]}}} = 0.4  and  d = 3.4.       <U>ANSWER</U>


<U>CHECK</U>

n       {{{a[n]}}}
----------------

1	0.4		
2	3.8		
3	7.2		
4	10.6		
5	14	sum from 1 to 5 = 	36       ! Correct !
6	17.4		
7	20.8		
8	24.2		
9	27.6	sum from 1 to 9 = 	126      ! Correct !
</pre>

Solved.


Notice that I solved the problem MENTALLY, without using equations.


It was my goal from the very beginning, and therefore I organized my arguments correspondingly.


It was how this problem was designed and was intended to be solved.


------------------


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/Marh-Olimpiad-level-problem-on-arithmetic-progression.lesson>Math Olimpiad level problem on arithmetic progression</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.