If for some positive integer n , the coefficients of three consecutive terms in the binomial expansion of 1 + x n + 5 are in the ratio 5 : 10 : 14 , then the largest coefficient in the expansion is :

Step-by-step solution

Let three consecutive terms are T r ,   T r + 1 ,   T r + 2 So, C r - 1 n + 5 C r n + 5 = 5 10 and C r n + 5 C r + 1 n + 5 = 10 14 Using property C r n C r - 1 n = n - r + 1 r , we have n + 5 - r + 1 r = 2 and n + 5 - r + 1 + 1 r + 1 = 7 5 ⇒ n - 3 r + 6 = 0 and 5 n - 12 r + 18 = 0 Solving both equations, n = 6 ,   r = 4 Hence, the greatest coefficient will be of middle term, which is = C 5 n + 5 = C 5 11 = 462