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?
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.
Hello, xwrathbringerx!
Another approach . . .
Let $\displaystyle r$ = common ratio.Insert 3 numbers between 3 and 48 such that the 5 numbers form a GP.
. . $\displaystyle \begin{array}{cc}\text{The 1st term is:}& 3 \\
\text{The 2nd term is:}&3r\\
\text{The 3rd term is:}&3r^2\\
\text{The 4th term is:}&3r^3 \\
\text{The 5th term is:}&3r^4 \end{array}$
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: .$\displaystyle \begin{array}{cc}
3,\: 6,\: 12,\: 24,\: 48 \\ \\[-3mm]
3,\text{-}6,12,\text{-}24,48 \end{array}$