Consider the expansion of (1+x)^ 2 n+1 If the coefficients of x^r and x^ r+1 are equal in the expansion, then r is equal to

Step-by-step solution

aligned & (1+x)^ 2 n+1 = ^ (2 n+1) C₀ x^0+ ^ (2 n+1) C₁ x^1 & + .+ (2 n+1) 2 n+1 (x)^ 2 n+1 aligned Coefficient of x ^ r = ^ (2 n +1) Cr Coefficient of x ^ r +1 = ^ (2 n +1) Cr +1 aligned & ^ (2 n+1) C_r= ^ (2 n+1) Cr +1 & (2 n+1)! r!(2 n+1-r)! = (2 n+1)! (r+1)!(2 n-r)! & (2 n-r)! (2 n+1-r)(2 n-r)! = r! (r+1) r! & (r+1)=2 n+1-r & r=n aligned