Question 1037823
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three numbers form a geometric progression if we double the middle number we get an arithmetic progression.
the common ratio of the geometric progression is ?
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The original geometric progression:   {{{a}}},  {{{ar}}},  {{{ar^2}}}.


The new sequence is:  {{{a}}},  {{{2ar}}},  {{{ar^2}}}.


The new progression is arithmetic.  It means that  {{{a[3]}}} - {{{a[2]}}} = {{{a[2]}}} - {{{a[1]}}}.


In other words,  {{{ar^2 - 2ar}}} = {{{2ar - a}}}.


Then  {{{a^2 - 4ar + a}}} = {{{0}}},  ---> 

{{{r^2 - 4r + 1}}} = {{{0}}},

{{{r[1]}}} = {{{1 + sqrt(3)}}}   (positive),  

{{{r[2]}}} = {{{1 - sqrt(3)}}}   (negative).


The previous solution was wrong.