about to only mathematics

Options

A. an even integer
B. an odd integer
C. the square of an integer
D. the cube of an integer

Answer

C

Explanation

Let a-3d, a-d, a+d, a+3d be four consecutive terms of an A.P. with integer terms. Clearly, the common difference is 2d.
Since the terms are integers, therefore a and d are also integers.
Also,
`(a-3d)(a-d)(a+d)(a+3d)+(2d)^(4)`
`=(a^(2)-9d^(2))(a^(2)-d^(2))+16d^(4)`
`=a^(4)-10a^(2)d^(2)+9d^(4)+16d^(4)`
`=a^(4)-10a^(2)d^(2)+25d^(4)`
`=(a^(2)-5d^(2))^(2)`, which is square of an integer as a nad d are integers.