a + a + a + a and ( – a ) = C ( ) – C ( ) ( a ) + C ( ) ( a ) – C ( ) ( a ) + C ( ) ( a ) – C ( a ) = – a + a – a + a – a Thus ( + a ) ( – a ) = ( + a + a + a + a ) ( – a + a – a + a – a ) The complete multiplication of the two brackets need not be carried out. We write only those terms which involve a . This can be done if we note that a r . a – r = a .
The terms containing a are ( a ) + ( a ) (– a ) + ( a ) ( a ) + ( a ) (– a ) + ( a ) ( ) = – a MATHEMATICS Thus, the coefficient of a in the given product is – . Example Find the r th term from the end in the expansion of ( x + a ) n . Solution There are ( n + ) terms in the expansion of ( x + a ) n . Observing the terms we can say that the first term from the end is the last term, i.e., ( n + ) th term of the expansion and n + = ( n + ) – ( – ).
The second term from the end is the n th term of the expansion, and n = ( n + ) – ( – ). The