If the coefficients of three consecutive terms in the expansion of (1 + x)^ n are in the ratio 1 : 2 : 3 , then the value of n is

Step-by-step solution

Let the three consecutive terms be the r^ th , (r+1)^ th , and (r+2)^ th terms. Their coefficients in the expansion of (1 + x)^ n are ^ n C_ r-1 , ^ n C_ r , and ^ n C_ r+1 respectively. We are given that ^ n C_ r-1 : ^ n C_ r : ^ n C_ r+1 = 1 : 2 : 3 . From the first two terms: ^ n C_ r ^ n C_ r-1 = 2 1 Using the property ^ n C_ r ^ n C_ r-1 = n-r+1 r , we have: n-r+1 r = 2 n - r + 1 = 2r n - 3r + 1 = 0 (1) From the next two terms: ^ n C_ r+1 ^ n C_ r = 3 2 Using the same property (replacing r with r+1 ): n-(r+1)+