Q1Short answer
+ 3 + 5 + ... + (2k – 1) + (2k + 1) = k2 + (2k + 1) (Why?) = k2 + 2k + 1 = (k + 1)2 Thus, P(k + 1) is true, whenever P(k) is true. Hence, by the Principle of Mathematical Induction, P(n) is true for all n ∈ N. n −1 n ( n − 1) ( n + 1)
+ 3 + 5 + ... + (2k – 1) + (2k + 1) = k2 + (2k + 1) (Why?) = k2 + 2k + 1 = (k + 1)2 Thus, P(k + 1) is true, whenever P(k) is true. Hence, by the Principle of Mathematical Induction, P(n) is true for all n ∈ N. n −1 n ( n − 1) ( n + 1)