Four different integers form an increasing A.P. such that one of them is the square of the remaining numbers. Find the largest number.

Let the A.P. bea−d, a, a+d, a+2d . Note that a and d must be integers. Also as this is an increasing a+2d is the largest. We have a+2d=(a−d)2+a2+(a+d)2                              =3a2+2d2 ⇒       3a2−a+2d2−2d=0 As a is real,  1−8(d2−d)≥0 ⇒         d2−d−18≤0 ⇒         (d−12)2≤38 ⇒         12−322≤d≤12+322 As d is an integer, d=0,1 But d≠0 , therefore, d=1 Thus 3a2−a=0⇒ a=0  or  a=1/3 As a is an integer,a=0 .Hence, required number is 2.