Question

Find a , b and n in the expansion of ( a + b ) n if the first three terms of the expansion are 729, 7290 and 30375, respectively.

Answer 1

The given expression is ( a+b ) n and its first three terms of the expansion are 729, 7290 and 30375 . We have to find the value s of a , b and n .

T r+1 = C n r a n−r ( b ) r

The first three terms of the expansion are given as 729 , 7290 and 30375 respectively.

Therefore, we obtain

T 1 =729 C n 0 a n−0 ( b ) 0 =729 a n =729 (1)

T 2 =7290 C n 1 a n−1 ( b ) 1 =7290 n a n−1 b 1 =7290 (2)

T 3 =30375 C n 2 a n−2 ( b ) 2 =30375 n( n−1 ) 2 a n−2 b 2 =30375 (3)

By dividing the equation (2) by (1), we get

n a n−1 b a n = 7290 729 nb a =10 (4)

By dividing the equation (3) by (2), we get

n( n−1 ) a n−2 b 2 2n a n−1 b = 30375 7290 ( n−1 )b 2a = 30375 7290 ( n−1 )b a = 30375×2 7290 = 25 3

nb a − b a = 25 3 10− b a = 25 3 [Using Equation 4]

b a =10− 25 3 = 5 3 (5)

From equation (4) and (5), we get

n× 5 3 =10 n=6

Substitute the value of n=6 in equation (1),

a 6 =729 a= 729 6 =3

From equation (5),

b a = 5 3 b 3 = 5 3 b=5

Thus, the values of a=3 , b=5 , and n=6 .