If the coefficients of mth,(m+1)th  and (m+2)th  terms in the expansion (1+x)n  are in A.P., then

Given expansion (1+x)n is,  We have general term in the expansion (x+a)n (∴  Tr+1= nCr xn−r (a)r  be the expansion of (x+a)n)  mth  term:  T(m−1)+1= nCm−1 xn−m+1 (a)m−1   The coefficient ofmth  term is nCm−1 (m+1)th  term:  Tm+1= nCm xn−m (a)m   The coefficient of(m+1)th  term is nCm (m+2)th  term: T(m+1)+1= nCm+1 xn−m−1 (a)m+1   The coefficient of(m+2)th  term is nCm+1 The coefficients of mth,(m+1)th  and (m+2)th  terms in the expansion (1+x)n  are in A.P., then(if a,b and c are in A.P then b=a+c2⇒2b=a+c )2. nCm= nCm−1+ nCm+1  (∴nCr=n!(n−r)! r!) ⇒2n!m!(n−m)!=n!(m−1)!(n−m+1)!+n!(m+1)!(n−m−1)! ⇒2n!(m−1)!m(n−m−1)!(n−m)=n!(m−1)!(n−m+1)!(n−m)(n−m+1)+n!m(m+1)(m−1)!(n−m−1)! ⇒2n!m(n−m)=n!(n−m)(n−m+1)+n!m(m+1) ⇒2m(n−m)=m(m+1)+(n−m)(n−m+1)(n−m)(n−m+1)m(m+1) ⇒2(n−m+1)(m+1)=m(m+1)+(n−m)(n−m+1) ⇒2(mn−m2+m+n−m+1)=m2+m+n2−nm+n−mn+m2−m ⇒(n−2m)2=n+2