If the coefficients of  P th, ( p + 1)th and (p + 2)th terms in the expansion of (1 + x)n are in AP, then

coefficients of pth , (p + 1)th and (p + z)th terms in the expansion (1 + x)" are  nCp−1, nCp′   nCp+1, respectively.Since, these are in AP.∴ 2nCp=nCp−1+nCp+1⇒2n!(n−p)!p!=n!(n−p+1)!(p−1)!+n!(n−p−1)!(p+1)! ⇒2(n−p)!p!=p(n−p+1)(n−p)!p!+n−p(n−p)!(p+1)p!        ⇒ 21=p(n−p+1)+n−pp+1⇒ n2−n(4p+1)+4p2−2=0