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Math Help - Geometric Progression.

  1. #1
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    Exclamation 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?
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  2. #2
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    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.
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  3. #3
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    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 \\<br />
\text{The 2nd term is:}&3r\\<br />
\text{The 3rd term is:}&3r^2\\<br />
\text{The 4th term is:}&3r^3 \\<br />
\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}<br />
3,\: 6,\: 12,\: 24,\: 48 \\ \\[-3mm]<br />
3,\text{-}6,12,\text{-}24,48 \end{array}

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