`" if " a_(a) ,a_(2), a_(3)….." are in A.P, then find the value of the following determinant:"`
`|{:(a_(p)+a_(p+m) +a_(p+2m),,2a_(p)+3a_(p+m)+4a_(p+2m),,4a_(p)+9a_(p+m)+16a_(p+2m)),(a_(p)+a_(q+m)+a_(q+2m),,2a_(q)+3a_(q+m) +4a_(q+2m),,4a_(q)+9a_(q+m)+16a_(q+2m)),(a_(r)+a_(r+m)+a_(r+2m),,2a_(r)+3a_(r+m)+4a_(r+2m),,4a_(r) +9a_(r+m)+16a_(r+2m)):}|`

Explanation

`|{:(a_(p)+a_(p+m) +a_(p+2m),,2a_(p)+3a_(p+m)+4a_(p+2m),,4a_(p)+9a_(p+m)+16a_(p+2m)),(a_(p)+a_(q+m)+a_(q+2m),,2a_(q)+3a_(q+m) +4a_(q+2m),,4a_(q)+9a_(q+m)+16a_(q+2m)),(a_(r)+a_(r+m)+a_(r+2m),,2a_(r)+3a_(r+m)+4a_(r+2m),,4a_(r) +9a_(r+m)+16a_(r+2m)):}|`
Applying `C_(2) to C_(2) -2C_(1) " and " C_(3) to C_(3) -4C_(1), ` we get
`|{:(a_(p)-a_(p+2m),,a_(p+m)+2a_(p+2m),,2a_(p+2m)),(a_(q)-a_(q+2m),,a_(q+m)+2a_(q+2m),,2a_(q+2m)),(a_(r) -a_(r+2m),,a_(r+m)+2a_(r+2m),,2a_(r+2m)):}|`
Applying `C_(2) to C_(2) -C_(3) to C_(1) +(1)/(2) C_(3)` and then taking 2 common from `C_(3)` we get
`=2 |{:(a_(p),,a_(p+m),,a_(p+2m)),(a_(q),,a_(q+m),,a_(p+2m)),(a_(r),,a_(r+m),,a_(r+2m)):}|`
Applying `C_(1) to C_(1)+C_(3)+2C_(2)`
`=2 |{:(a_(p)+a_(p+2m),,a_(p+m),,a_(p+2m)),(a_(q)+a_(q+2m),,a_(q+m),,a_(p+2m)),(a_(r)+a_(r+2m),,a_(r+m),,a_(r+2m)):}|`
`=2 |{:(0,,a_(p+m),,a_(p+2m)),(0,,a_(q+m),,a_(p+2m)),(0,,a_(r+m),,a_(r+2m)):}|`
`(Ax a_(p)a_(p+m),a_(p+2m)` are in A.P.)
`=0`