occurs in the ( r + ) th term of the expansion ( x + y ) . Now T r + = C r x – r ( y ) r = C r r . x – r . y r .
Comparing the indices of x as well as y in x y and in T r + , we get r = . Thus, the coefficient of x y is C = ! ! !
Example The second, third and fourth terms in the binomial expansion ( x + a ) n are , and , respectively. Find x , a and n . Solution Given that second term T = MATHEMATICS We have T = n C x n – . a So n C x n – .
a = ... ( ) Similarly n C x n – a = ... ( ) and n C x n – a = ... ( ) Dividing ( ) by ( ), we get C C i.e., )!
)! or = n ... ( ) Dividing ( ) by ( ), we have ( n ... ( ) From ( ) and ( ), ( ) Thus, n = Hence, from ( ), x a = , and from ( ), Solving these equations for a and x , we get x = and a = .
Example The coefficients of three consecutive terms in the expansion of ( + a ) n are in the ratio1: : . Find n . Solution Suppose the three consecutive terms in the expansion of ( + a ) n are ( r – ) th , r th and ( r + ) th terms. The ( r – ) th term is n C r – a r – , and its