Common ratio problems

Hello, nightrider456!

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

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

Quote:

Solve for the variable and find the common ratio.

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

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

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

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

Quote:

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

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

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

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

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

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