Question 1062762
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Three numbers are in AP such that their sum is 18 and sum of their squares is 158 the greatest number among them is
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<pre>
Let x be the middle term of our 3-term AP.

Then {{{a[1]}}} = x-d, {{{a[2]}}} = x and {{{a[3]}}} = x + d, and the sum of the three terms is  (x-d) + x + (x+d) = 3d.

So we have 3x = 18 and x = {{{18/3}}} = 6.


Next, the square of the first term is {{{(x-d)^2}}} = {{{x^2 - 2dx + d^2}}},
      the square of the middle term is {{{x^2}}} and
      the square of the third term is {{{(x+d)^2}}} = {{{x^2 + 2dx + d^2}}}.

Add these tree squares, and you will get their sum as {{{3x^2 + 2d^2}}}.
Now recall that x = 6, hence, x^2 = 36, and the equation for the squares becomes

{{{3*36 + 2d^2}}} = 158.

--->  {{{2d^2}}} = 158-108 = 50  --->  {{{d^2}}} = {{{50/2}}} = 25 --->  d = +/- 5.


So, there are two AP progressions satisfying the condition:

1)  6-5 = 1, 6,  6+5 = 11,   and  the reversed sequence


2)  11, 6, 5.
</pre>

<U>Answer</U>. The greatest number of the three is 11.



For similar solved problems see the lesson 

&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> 

in this site.



On arithmetic progressions, there is a bunch of lessons

&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/Chocolate-bars-and-arithmetic-progressions.lesson>Chocolate bars and 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, 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>.