Math Help - Geometric Progression.

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 $3,x,y,z,48$

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

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

By (1) and (2),

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

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

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

Hope this helps.

3. Hello, xwrathbringerx!

Another approach . . .

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

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

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

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

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