Question 1004512
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Three numbers whose sum is 3 form an arithmetic sequence and their squares form a geometric sequence. 
What are the numbers?
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<U>Answer</U>. The numbers are &nbsp;{{{1 - sqrt(2)}}}, &nbsp;{{{1}}}, &nbsp;and {{{1 + sqrt(2)}}}.


<U>Solution</U>


Since three numbers form an arithmetic progression, we can represent them as &nbsp;(a-d), &nbsp;a, &nbsp;and &nbsp;(a+d), 

where &nbsp;a &nbsp;is the middle term and &nbsp;d&nbsp; is the common difference.


The squares of these numbers are &nbsp;{{{(a-d)^2}}}, &nbsp;{{{a^2}}}, &nbsp;and &nbsp;{{{(a+d)^2}}}.

The fact that the squares form a geometric progression means that 


{{{((a+d)^2)/a^2}}} = {{{a^2/((a-d)^2)}}} &nbsp;&nbsp;&nbsp;&nbsp;(1)


(the ratio of the third to the second is equal to the ratio of the second to the first, as these ratios are 

the common ratio of the geometric progression). &nbsp;The formula &nbsp;(1) implies, &nbsp;after simplifying, that


{{{d^2}}} = {{{2a^2}}}, &nbsp;&nbsp;&nbsp;&nbsp;or &nbsp;&nbsp;&nbsp;&nbsp;d = a{{{sqrt(2)}}}.


Next, &nbsp;since the sum of the tree numbers is &nbsp;3, &nbsp;we conclude that


(a-d) + a + (a+d) = 3a = 3, &nbsp;&nbsp;and, &nbsp;&nbsp;hence, &nbsp;&nbsp;a = 1. &nbsp;&nbsp;In turn, it means that &nbsp;d = {{{sqrt(2)}}}.


Thus our sequence of three numbers is &nbsp;&nbsp;{{{1 - sqrt(2)}}}, &nbsp;1, &nbsp;and &nbsp;{{{1 + sqrt(2)}}}.


As a check, &nbsp;it is clear that the found numbers form arithmetic progression and their sum is &nbsp;3.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Their squares are &nbsp;{{{3-2sqrt(2)}}}, &nbsp;1, &nbsp;and &nbsp;{{{3+2sqrt(2)}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The consequtive ratios of the squares are &nbsp;&nbsp;{{{1/(3-2sqrt(2))}}} &nbsp;&nbsp;and &nbsp;&nbsp;{{{(3+2sqrt(2))/1}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Finally, &nbsp;you can check yourself that these ratios are equal, &nbsp;which means that the squares form 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the geometric progression. 


The solution is completed.


Thank you for submitting the fresh, &nbsp;sweet and crispy problem!