If a, b, c are in A.P. and p is the A.M. between a and b  and q is the A.M. between b and c, then

∵a,b,c are AP., ∴ 2b=a+c                 (1)∵ p is the A.M. between a and b∵ p=a+b2                    (2)∵ q is the A.M. between b and c∵     q=b+c2                 (3)Adding Eq. (2) and (3)p+q=a+b2+b+c2=a+c+2b2=2b+2b2=2b  [Using (1)]∴ 2b=p+q or b=p+q2Hence, b is the A.M. between p and q.