In a set of four numbers, the first three are in G.P. and the last three are in A.P. with difference 6. If the first number is the same as the fourth, find the four numbers.

Explanation

Let the last three numbers in AP, be `a,a+6,a+12` .
`" " " " [:." 6 is the common difference "]`
If first number is b, then four numbes are
`b,a,a+6,a+12`
But given, `b=a+12`
`:.` Four numbers are `a+12,a,a+6,a+12" " ".....(i)"`
Since, first three numbers are in GP.
Then, `a^(2)=(a+12)(a+6)`
`implies a^(2)=a^(2)+18a+72`
`implies 18a+71=0`
`:. " " a=-4 " " [" from Eq.(i) "]`
Hence, four numbers are `8,-4,2,8` .