( r – ) th and ( r – ) th terms in the expansion of ( + x ) are equal, find r . Solution The coefficients of ( r – ) th and ( r – ) th terms of the expansion ( + x ) are C r – and C r – , respectively. Since they are equal so C r – = C r – Therefore, either r – = r – or r – = – ( r – ) [Using the fact that if n C r = n C p , then either r = p or r = n – p ] So, we get r = – or r = . r being a natural number, r = – is not possible.
So, r = . Miscellaneous Exercise on Chapter . Find a , b and n in the expansion of ( a + b ) n if the first three terms of the expansion are , and 30375, respectively. .
Find a if the coefficients of x and x in the expansion of ( + ax ) are equal. . Find the coefficient of x in the product ( + x ) ( – x ) using binomial theorem. .
If a and b are distinct integers, prove that a – b is a factor of a n – b n , whenever n is a positive integer. [ Hint write a n = ( a – b + b ) n and expand] .