Question 1103432
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<pre>
Let  "r" be the unknown common term.

Then

S = {{{11 + 11*r + 11*r^2 + 11*r^3 + ellipsis + 11*r^11}}} = 2922920,  which implies

1 + r + r^2 + r^3 + . . . + r^11 = {{{2922920/11}}} = 265720,   or


{{{(r^12-1)/(r-1)}}} = 265720


You can check that r= 3 is the solution to this equation:  {{{(3^12-1)/(3-1)}}} = 265720.
</pre>

<U>Answer</U>.  The common ratio of this progression is 3.


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<pre>
It is not difficult to prove that r= 3 is the UNIQUE solution to the problem.


Indeed, if r >=1 then the sum  1 + r + r^2 + . . . + r^11 is monotonic function of r.


Next, if  0 < r < 1, then this sum is less than 12.


Further, if  -1 <= r < 0, then AGAIN this sum is less than 12.


Finally, if  r < -1,  then  {{{(r^12-1)/(r-1)}}}  has POSITIVE numerator and negative denominator, which means that this rational function is NEGATIVE.


The plot below ILLUSTRATES this behavior of the function  1 + r + r^2 + . . . + r^11.



{{{graph( 330, 330, -6.5, 3.5, -30000, 300000,
          (x^12-1)/(x-1), 265720
)}}}


Plot y = {{{(x^12-1)/(x-1)}}} (red)  and y = 265720 (green)
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