The question I am asking is this
Suppose $a$, $b$, and $c$ are real numbers such that $a$, $b$, $c$ and $a$, $b + 1$, $c + 2$ are both geometric sequences in their respective orders. Compute the smallest possible value of $(a + b + c)^2$.
This problem is related to a question in an AMC competition. I have nowhere to start except $xa=b$, $xb= c$, and $ya=b+1$, $y(b+1)=c+2$.
Any help on this?
Hint $a c = b^2$ and $a (c+2) = (b+1)^2$. Compute $b$ and $c$ in terms of $a$.