Question

Find a if the coefficients of x 2 and x 3 in the expansion of (3 + ax ) 9 are equal.

Answer 1

The given expression is ( 3+ax ) 9 , and the coefficients of x 2 and x 3 are equal, then find value of a ,

T r+1 = C n r a n−r ( b ) r

Suppose x 2 occurs in ( r+1 ) th term of the expansion ( 3+ax ) 9 ,

T r+1 = C 9 r ( 3 ) 9−r ( ax ) r = C 9 r ( 3 ) 9−r a r x r

Comparing the indices of x in x 2 and in T r+1 ,

r=2

Thus, the coefficient of x 2 is

C 9 2 ( 3 ) 9−2 ( a ) 2 = 9! 2!7! ( 3 ) 7 a 2 =36 ( 3 ) 7 a 2

Assume x 3 occurs in ( k+1 ) th term in the expansion of ( 3+ax ) 9 ,

T k+1 = C 9 k ( 3 ) 9−k ( ax ) k = C 9 k ( 3 ) 9−k a k x k

Comparing the indices of x in x 3 and in T k+1 ,

k=3

Thus the coefficient of x 3 is

C 9 3 ( 3 ) 9−3 ( a ) 3 = 9! 3!6! ( 3 ) 6 a 3 =84 ( 3 ) 6 a 3

It is given that the coefficients of x 2 and x 3 are the same. Therefore,

84 ( 3 ) 6 a 3 =36 ( 3 ) 7 a 2 84a=36×3 a= 36×3 84 a= 9 7

Thus, the required value of a is 9 7 .