Let M = 0 - α α 0 , where α is a non-zero real number and N = ∑ k = 1 49 M 2 k . If I - M 2 N = - 2 I , then the positive integral value of α is ______.

Step-by-step solution

Given, M = 0 - α α 0 Now, M 2 = - α 2 0 0 - α 2 = - α 2 I Also given, N = ∑ k = 1 49 M 2 k ⇒ N = M 2 + M 4 + … … . + M 98 = - α 2 + α 4 - α 6 + … . . I = - α 2 1 - - α 2 49 1 + α 2 . I ⇒ N 1 + α 2 = - α 2 α 98 + 1 ⇒ N I - M 2 = - α 2 α 98 + 1 Now comparing with I - M 2 N = - 2 I We get, α = 1 .