Question

Find n , if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of

Answer 1

The given expression is ( 2 4 + 1 3 4 ) n , and the ratio of fifth term from the beginning to the fifth term from the end is equal to 6 :1 .

Fifth term from the beginning

= C n 4 a n−4 b 4

Fifth term from the end

= C n n−4 a 4 b n−4

Therefore, the expansion of ( 2 4 + 1 3 4 ) n has fifth term from the beginning as C n 4 2 4 n−4 ( 1 3 4 ) 4

And the fifth term from the end as C n n−4 2 4 4 ( 1 3 4 ) n−4 .

C n 4 2 4 n−4 ( 1 3 4 ) 4 = C n 4 ( 2 4 ) n ( 2 4 ) 4 × 1 3 = C n 4 ( 2 4 ) n 2 × 1 3 = n! 6×4!( n−4 )! ( 2 4 ) n (1)

C n n−4 2 4 4 ( 1 3 4 ) n−4 = C n n−4 ( 3 4 ) 4 ( 3 4 ) n ×2 = C n n−4 3 ( 3 4 ) n ×2 = 6n! 4!( n−4 )! 1 ( 3 4 ) n (2)

Ratio is given in the question as 6 :1 .So, from equation (1) and (2), we get

n! 6×4!( n−4 )! ( 2 4 ) n : 6n! 4!( n−4 )! 1 ( 3 4 ) n = 6 :1 ( 2 4 ) n 6 : 6 ( 3 4 ) n = 6 :1 ( 2 4 ) n 6 ÷ 6 ( 3 4 ) n = 6 ÷1 ( 2 4 ) n 6 × ( 3 4 ) n 6 = 6 ×1 ( 6 4 ) n =36 6 6 n 4 = 6 2+ 1 2 n 4 = 5 2 n=4× 5 2 =10

Thus, the value of n=10 for the expression ( 2 4 + 1 3 4 ) n .