`" if " a_(1) ,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

To solve the given determinant, we will use the properties of arithmetic progression (A.P.) and the properties of determinants.

### Step-by-Step Solution:

1. **Understanding the Terms in A.P.**:
Given that \( a_1, a_2, a_3, \ldots \) are in A.P., we can express:
- \( a_{p+m} = a_p + m \cdot d \)
- \( a_{p+2m} = a_p + 2m \cdot d \)
...