Find three numbers in G.P. whose sum is 52 and the sum of whose products i pairs is 624.

Explanation

Let the required numbers be `a, ar, ar^(2)`. Then,
`(a+ar+ar^(2))=52 rArr a(1+r+r^(2))=52` ...(i)
and `a.ar+ar.ar^(2)+a.ar^(2)=624 rArr a^(2)r(1+r+r^(2))=624` ...(ii)
On dividing (ii) by (i), we get
`ar=12` and therefore, `a=12/r`.
Putting `a=12/r` in (i), we get
`12/r. (1+r+r^(2))=52 rArr 3(1+r+r^(2))=13r`
`rArr 3r^(2)-10 r+3=0`
`rArr (3r-1) (r-3)=0`
`rArr r=1/3 or r=3`.
`:. a=36 or a=4`.
Hence, the required numbers are 36, 12, 4 or 4, 12, 36.