Question 1151377
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We are given 


    {{{a[3]}}} = {{{r^2*a[1]}}} = {{{9*a[1]}}}.


It implies  {{{r^2}}} = 9;  hence, taking the square root of both sides,

we have TWO possible values for the common ratio "r"  :  r = 3  or  r = -3.


If r= 3, then the sum of the first four terms is  {{{a[1] + 3*a[1] + 3^2*a[1] + 3^3*a[1]}}} = {{{(1 + 3 + 9 + 27)*a[1]}}} = {{{40*a[1]}}}.


    Hence, in this case  K = 40.



If r= -3, then the sum of the first four terms is  {{{a[1] - 3*a[1] + (-3)^2*a[1] + (-3)^3*a[1]}}} = {{{(1 - 3 + 9 - 27)*a[1]}}} = {{{-20*a[1]}}}.


    Hence, in this case  K = -20.



<U>ANSWER</U>.  Two values are possible for K:  40 and - 20.
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Solved.