1. ## Common ratio problems

Been having problems with the solutions to the following problems dealing with finding the common ratio.

Solve for the missing variable and find the common ratio
1.)$\displaystyle \sqrt{x},3,3\sqrt{x}$

2.)$\displaystyle m+2,m+4,2m+11$

2. Hello, nightrider456!

If we are given three terms $\displaystyle a,b,c$ with a common ratio,

. . then: .$\displaystyle \frac{a}{b} \:=\:\frac{b}{c}$

Solve for the variable and find the common ratio.

. . $\displaystyle (1)\;\;\sqrt{x},\;3,\;3\sqrt{x}$

We have: .$\displaystyle \frac{\sqrt{x}}{3} \;=\;\frac{3}{3\sqrt{x}} \quad\Rightarrow\quad 3x \:=\:9 \quad\Rightarrow\quad x \:=\:3$

The three terms are: .$\displaystyle \sqrt{3},\;3,\;3\sqrt{3}$

. . The common ratio is: .$\displaystyle \sqrt{3}$

$\displaystyle (2)\;\;m+2,\;m+4,\;2m+11$

We have: .$\displaystyle \frac{m+2}{m+4} \:=\:\frac{m+4}{2m+11} \quad\Rightarrow\quad 2m^2 + 15m + 22 \:=\:m^2 + 8m + 16$

. . $\displaystyle m^2 + 7m + 6 \:=\:0 \quad\Rightarrow\quad (m+1)(m+6) \:=\:0$

Hence: .$\displaystyle m \;=\;-1,\:-6$ . . . two solutions

If $\displaystyle m = -1$, the three terms are: .$\displaystyle 1,\:3,\:9$
. . The common ratio is: .$\displaystyle 3$

If $\displaystyle m = -6$, the three terms are: .$\displaystyle -4,\:-2,\:-1$
. . The common ratio is: .$\displaystyle \tfrac{1}{2}$