1 + 3 + 5 + ... + (2n – 1) = n^2

Question

Prove statement, by using the Principle of Mathematical Induction for all n ∈ N, that :

1 + 3 + 5 + ... + (2n – 1) = n²

Answer 1

Let the given statement P(n) be defined as P(n) : 1 + 3 + 5 +...+ (2n – 1) = n² , for n ∈ N. Note that P(1) is true, since 

P(1) : 1 = 1² 

Assume that P(k) is true for some k ∈ N, i.e., 

P(k) : 1 + 3 + 5 + ... + (2k – 1) = k² 

Now, to prove that P(k + 1) is true, we have 

1 + 3 + 5 + ... + (2k – 1) + (2k + 1) 

= k² + (2k + 1)

= k² + 2k + 1 = (k + 1)

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.