Let for the 9 th term in the binomial expansion of 3 + 6 x n , in the increasing powers of 6 x , to be the greatest for x = 3 2 , the least value of n is n 0 . If k is the ratio of the coefficient of x 6 to the coefficient of x 3 , then k + n 0 is equal to

Step-by-step solution

Given, 3 + 6 x n = 3 n 1 + 2 x n Now if T 9 is numerically greatest term, then T 8 ≤ T 9 ≥ T 10 So, C 7 n 3 n - 7 6 x 7 ≤ C 8 n 3 n - 8 6 x 8 ≥ C 9 n 3 n - 9 6 x 9 ⇒    n ! n - 7 ! 7 ! 9 ≤ n ! n - 8 ! 8 ! 3 · 6 x ≥ n ! n - 9 ! 9 ! 6 x 2 ⇒ 9 n - 7 n - 8 ⏟ ≤ 18 3 2 n - 8 8 ≥ 36 9 . 8 9 4 ⏟ ⇒ 72 ≤ 27 n - 7 and 27 ≥ 9 n - 8 ⇒ 29 3 ≤ n and n ≤ 11 So, n 0 = 10 For 3 + 6 x 10 Now T r + 1 = C r 10 3 10