The terms `a_1, a_2, a_3` from an arithmetic sequence whose sum s 18. The terms `a_1+1,a_2, a_3,+2,` in that order, form a geometric sequence. Then the absolute value of the sum of all possible common difference of the A.P. is ________.

Answer

(-1)

Explanation

Let `a_(1)=a-d,a_(2)=a,a_(3)=a+d`
`therefore3a=18` or a=6
Hence, the number in A.P.
`6-d,d,6+d`
`a_(1)+1,a_(2),a_(3)+2` in G.P.
i.e., 7-d,6,8+d in G.P.
`therefore` 36=(7-d)(8+d)
`36=56-d-d^(2)`
`d^(2)+d-20=0`
Hence, the sum of all possible common different is -1.