r + ) BINOMIAL THEOREM or n – nr – n + r – = i.e., n – n ( r + ) + r – = Example Show that the coefficient of the middle term in the expansion of ( + x ) n is equal to the sum of the coefficients of two middle terms in the expansion of ( + x ) n – . Solution As n is even so the expansion ( + x ) n has only one middle term which is th i.e., ( n + ) th term. The ( n + ) th term is n C n x n . The coefficient of x n is n C n Similarly, ( n – ) being odd, the other expansion has two middle terms, th n −+ and th n −+ i.e., n th and ( n + ) th terms.
The coefficients of these terms are n – C n – and n – C n , respectively. Now n – C n – + n – C n = n C n [As n C r – + n C r = n + C r ]. as required. Example Find the coefficient of a in the product ( + a ) ( – a ) using binomial theorem.
Solution We first expand each of the factors of the given product using Binomial Theorem. We have ( + a ) = C + C ( a ) + C ( a ) + C ( a ) + C ( a ) = + ( a ) + ( a ) + ( a ) + a . = +