th and th terms of the expansion ( + a ) are equal. Solution The ( r + ) th term of the expansion ( x + y ) n is given by T r + = n C r x n – r y r . For the th term, we have, r + = , i.e., r = Therefore, T = T + = C ( ) – a = C a . Similarly, T = C a Given that T = T So C ( ) a = C ( ) a Therefore C C i.e., a = C C × !
! × × = Example Show that the middle term in the expansion of ( + x ) n is ( ) ! . .
... n n x n , where n is a positive integer. BINOMIAL THEOREM Solution As n is even, the middle term of the expansion ( + x ) n is th , i.e., ( n + ) th term which is given by, T n + = n C n ( ) n – n ( x ) n = n C n x n = ( )! !
Example Find the coefficient of x y in the expansion of ( x + y ) . Solution Suppose x y