The question goes like this , The sum of first three terms of a GP is 13/12 and their product is -1 Find the common ratio and terms of the GP
My answer went something like this : Assume 3 numbers to be a , ar , ar^2 where a is the first term and r is the common ratio Now as per question a^3.r^3=-1 and hence a=-1/r When I substitute this in the first condition i get: a(1+r+r^2)=13/12 after simplification -1/r- 1- 1/r=13/12 then I get r=-24/25 and terms to be 24/25 , -1 , 25/24
Here is how my book answer went They assumed the terms to be a/r then a then ar So product yielded them a^3=-1 and a=-1 and r was -3/4 or -4/3 after solving the Condition
Now why is my answer and their answer not the same ? Is it because cancelling r rules out the possibilities of including 0 to the term but they have also done the same thing cancelling the r to get a value in first place, or is the book at fault for not considering this answer. I am very perplexed
Your strategy for finding the answer is just as good as the one in the book. Your answer is not the same because, at some point, you simplified an expression incorrectly.
If $a = -\frac1r$ and $a(1 + r + r^2) = \frac{13}{12}$, then after substituting, we should get $$ -\frac1r(1+r+r^2) = \frac{13}{12} \implies -\frac1r - 1 - r = \frac{13}{12} $$ which still has the same solutions $r = -\frac34$ and $r = - \frac43$. Getting $-\frac1r - 1 - \frac1r = \frac{13}{12}$ instead is where your solution goes wrong.
We can also check that $\frac{24}{25},-1,\frac{25}{24}$ has sum $\frac{601}{600}$, not $\frac{13}{12}$.