1. ## Geometric Progression.

Hi

Insert 3 numbers between 3 and 48 such that the 5 numbers form a GP.

I only know that the common ratio is 2 by guess and check How do I do this properly?

2. Dear xwrathbringerx,

Suppose the geometrical progression is $\displaystyle 3,x,y,z,48$

Then $\displaystyle \frac{48}{z}=\frac{x}{3}\Rightarrow{xz=48\times{3} }$--------(1)

Also, $\displaystyle \frac{z}{y}=\frac{y}{x}\Rightarrow{y^2=xz}$----------(2)

By (1) and (2),

$\displaystyle y^2=48\times{3}\Rightarrow{y=\pm{12}}$

Since, $\displaystyle \frac{x}{3}=\frac{\pm{12}}{x}\Rightarrow{x=6}$

$\displaystyle \frac{z}{y}=\frac{y}{x}\Rightarrow{z=\frac{144}{6} =24}$

Hope this helps.

3. Hello, xwrathbringerx!

Another approach . . .

Insert 3 numbers between 3 and 48 such that the 5 numbers form a GP.
Let $\displaystyle r$ = common ratio.

. . $\displaystyle \begin{array}{cc}\text{The 1st term is:}& 3 \\ \text{The 2nd term is:}&3r\\ \text{The 3rd term is:}&3r^2\\ \text{The 4th term is:}&3r^3 \\ \text{The 5th term is:}&3r^4 \end{array}$

But we know that the $\displaystyle 5^{th}$ term is 48.

Hence, we have: .$\displaystyle 3r^4 \:=\:48 \quad\Rightarrow\quad r^4 \:=\:16 \quad\Rightarrow\quad r \:=\:\pm2$

There are two possible GP's: .$\displaystyle \begin{array}{cc} 3,\: 6,\: 12,\: 24,\: 48 \\ \\[-3mm] 3,\text{-}6,12,\text{-}24,48 \end{array}$