Question

In the binomial expansion of `(root(3)(2) + 1/root(3)(3))^n`, the ratio of the 7^th term from the beginning to the 7^th term from the end is 1:6; n is ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

In the binomial expansion of `(root(3)(2) + 1/root(3)(3))^n`, the ratio of the 7^th term from the beginning to the 7^th term from the end is 1:6; n is 9.00.

Explanation:

`(root(3)(2) + 1/root(3)(3))^n`

7^th term from the beginning = `""^nC_6 1/(3^(1/3))^6 (2^(1/3))^(n-6)`

7^th term from the end = (n – 7 + 2)th term from the beginning i.e. (n – 5)th term.

Hence, `(""^nC_6 1/(3^(1/3))^6 (2^(1/3))^(n-6))/(""^nC_(n-6) 1/((3^(1/3))^(n-6)) xx (2^(1/3))^6) = 1/6`

⇒ `(3^(1/3))^(n-12) xx (2^(1/3))^(n-12)` = 6^–1

⇒ `(6^(1/3))^(n-12)` = 6^–1

⇒ `(6)^((n-12)/3)` = 6^–1

⇒ `(n - 12)/3` = –1

⇒ n – 12 = –3

⇒ n = 9