Question 1083553
.
The arithmetic progression consists of 15 terms. If the sum of the 3 terms
in the middle is 27 and the sum 
of the last 3 terms is 18, then what is the sum of the first 3 terms?
~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
"If the sum of the 3 terms in the middle is 27" then {{{a[8]}}} = {{{27/3}}} = 9  (why ?)


Also, "if the sum of the last 3 terms is 18" then {{{a[14]}}} = {{{18/3}}} = 6.


Thus you have 

{{{a[8]}}} = {{{a[1]+7*d}}} = 9     (1)   and

{{{a[14]}}} = {{{a[1]+13*d}}} = 6   (2) 


It implies that 6d = 6 - 9 = -3.   Hence,  d = {{{(-3)/6}}} = -0.5.


Then {{{a[2]}}} = {{{a[8]-6d}}} = 9 - 6*(-0.5) = 9 + 3 = 12.


Hence, the sum  {{{a[1]+a[2]+a[3]}}} = 36    (why ?)
</pre>

<U>Answer</U>.  36.


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/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/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> 



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>.