Question 1134407
.
<pre>
The four terms of the given AP are nicely (symmetrically) located around their central point x= 5, which is their arithmetic mean  {{{20/4}}}.

I will use this fact in order for to simplify the solution. I also introduce  2d as the common difference of the AP 
(instead of using the tradition designation d for the common difference).


Then obviously


    {{{a[1]}}} = 5 - 3d,

    {{{a[2]}}} = 5 - d,

    {{{a[3]}}} = 5 + d,

    {{{a[4]}}} = 5 + 3d.


The sum of squares of these four terms is


    {{{a[1]^2 + a[2]^2 + a[3]^2 + a[4]^2}}} = {{{(25 - 30d + 9d^2)}}} + {{{(25 - 10d + d^2)}}} + {{{(25 + 10d + 9d^2)}}} + {{{(25 + 30d + d^2)}}} = {{{100 + 20d^2}}}.


So, for "d" you have this equation


    100 + 20d^2 = 120,


from which you get


    d^2 = {{{(120-100)/20}}} = 1;   hence,  d = +/- 1.


Thus the four terms of the AP are  5-3 = 2;  5-1 = 4;  5+1 = 6  and  5+3 = 8.


<U>ANSWER</U>.   The four terms of the AP are  2, 4, 6, 8.

          The reversed sequence  8, 6, 4, 2  is the solution, also.  It corresponds to the value  d= -1.
</pre>

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.



/\/\/\/\/\/\/\/



I want to remind you that as an educated visitor, you MUST express your thanks to EACH the solution

that I produce in response to your request.



I forced to make this reminder because MANY (if not all) visitors IGNORE THIS RULE.  Unfortunately.


Which I consider as an extreme degree of discourtesy.