Question 1190567
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Please help: the sum of the first two terms of a G.P is x.The sum of the last two is 1. 
if there are n term in all, calculate the common ratio
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<pre>
Let the GP be a, ar, ar^2, . . . , ar^(n-1).


The first term is "a", the common ratio is "r"; the number of terms is "n".


We are given 

    (a)  the value of n;            (the number of terms);

    (b)  a + ar = x;                (the sum of the 1st and the 2nd terms, x)

    (c)  ar^(n-2) + ar^(n-1) = 1.   (the sum of the two last terms, 1).



From (a), we have

    a*(1+r) = x.               (1)


From (b), we have

    {{{a*r^(n-2)*(1+r)}}} = 1.      (2)


Now divide equation (2) by equation (1). You will get, after canceling common factors

    {{{r^(n-2)}}} = {{{1/x}}}.


It implies  


    r = {{{(1/x)^(1/(n-2))}}} =  {{{root(n-2,1/x)}}} = {{{1/root(n-2,x)}}}.       <U>ANSWER</U>


Any of the equivalent forms in the line above is the answer.


So, the value of the common ratio "r" is  {{{highlight(UNIQUELY)}}}  {{{highlight(DETERMINED)}}}  by the input data x and n.
</pre>

Solved.


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<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As a conclusion</H3>

The problem is posed correctly.  &nbsp;It has &nbsp;(it admits) &nbsp;a unique solution.


Tutor @math_tutor2020 treated the problem incorrectly. &nbsp;He missed part of input data.


Therefore, &nbsp;his solution is incorrect.


His diagnosis 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;"there are infinitely many possible solutions here. 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;There isn't enough info to answer this current question."


is incorrect.


The problem is posed correctly, &nbsp;and I gave a complete solution/analysis.