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  4. The A.M. between two positive numbers exceeds the GM by 5 and the GM exceeds the H.M. by 4 . Then the numbers are-

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Q. The A.M. between two positive numbers exceeds the GM by 5 and the GM exceeds the H.M. by 4 . Then the numbers are-

Sequences and Series

Solution:

Let the two positive numbers are $a$ and $b$
By given condition
$\text { A.M. }=\text { GM. }+5 \text { or } A=G+5$..........(i)
$\text { G.M. }=\text { HM. }+4 \text { or } G=H+4$.......(ii)
we know $G ^2= AH$
or $G ^2=( G +5)( G -4) $ [By (i) and (ii)]
$\Rightarrow G=20$
$\therefore A =25$ and $H =16$
Again $A =\frac{ a + b }{2} $
$\therefore \frac{ a + b }{2}=25$
$ \Rightarrow a + b =50$
$G=\sqrt{a b}$
$\therefore \sqrt{a b}=20 $
$\Rightarrow a b=400$
solving the equation we get $a=10, b=40$