Question

Find three numbers in G.P. whose sum is 13 and the sum of whose squares is 91.

Answer 1

Let the required numbers be `a/r, a` and `ar`. Then,
`a/r+a+ar=13` ...(i)
and `a^(2)/r^(2)+a^(2)+a^(2)r^(2)=91`...(ii)
On squaring both sides of (i), we get
`(a/r+a+ar)^(2)=169`
`rArr (a^(2)/r^(2)+a^(2)+a^(2)r^(2))+2 (a^(2)/r+a^(2)+a^(2)r)=169` [using (ii)]
`rArr 91+26a=169` [using (i)]
`rArr 26a=78 rArr a=3`.
Putting `a=3` in (i), we get
`3/r+3+3r=13 rArr 3/r +3r=10 rArr 3r^(2)-10r+3=0`
`rArr (r-3)(3r-1)=0rArr r=3 or r=1/3`.
Hence, the required numbers are 1, 3, 9 or 9, 3, 1.