If some three consecutive coefficients in the binomial expansion of x + 1 n in powers of x are in the ratio 2 : 15 : 70 , then the average of these three coefficients is:

Step-by-step solution

From the given condition   n C r : n C r + 1 : n C r + 2 = 2 : 15 : 70 ⇒   n C r   n C r + 1 = 2 15 and   n C r + 1   n C r + 2 = 15 70 ⇒   n ! n - r ! · r ! n ! n - r - 1 ! · r + 1 ! = 2 15 and   n ! n - r - 1 ! · r + 1 ! n ! n - r - 2 ! · r + 2 ! = 15 70 ⇒ n - r - 1 ! · r + 1 ! n - r ! · r ! = 2 15 and n - r - 2 ! · r + 2 ! n - r - 1 ! · r + 1 ! = 3 14 ⇒ n - r - 1 ! · r + 1 · r ! n - r · n - r - 1 ! &