Let T_k denote the coefficient of x^k in the binomial expansion of (1+x)^n . If T_ r-1 + T_r = 84 and T_ r-1 + 2T_r + T_ r+1 = 210 , then the value of n + r is

Step-by-step solution

Given T_k = ,^ n C_ k . First equation: T_ r-1 + T_r = ,^ n C_ r-1 + ,^ n C_ r = ,^ n+1 C_ r = 84 Second equation: T_ r-1 + 2T_r + T_ r+1 = (T_ r-1 + T_r) + (T_r + T_ r+1 ) = ,^ n+1 C_ r + ,^ n+1 C_ r+1 = ,^ n+2 C_ r+1 = 210 Taking the ratio of the two simplified expressions: ,^ n+2 C_ r+1 ,^ n+1 C_ r = 210 84 Using the property ,^ N C_ R ,^ N-1 C_ R-1 = N R , we get: n+2 r+1 = 5 2 Cross-multiplying yields: 2n + 4 = 5r + 5 2n - 5r = 1 We also know ,^ n+1 C_ r = 84 . Testing small integer values for r in 2n = 5r + 1