Let T(n) be the statement: 1 + 3 + 5 + ... + (2n – 1) = n2
Basic Step: For n = 1, LHS = 1, RHS = 12
⇒ LHS = RHS ⇒ T(1) is true
Induction Step: Assume that T(k) is true, i.e., 1 + 3 + 5 + ... + (2k – 1) = k2
To obtain T (k + 1), add the (k + 1)th term
= 2 (k + 1) – 1 = 2k + 2 – 1 = 2k + 1 to both the sides.
Then, 1 + 3 + 5 + ... + (2k – 1) + (2k + 1) = k2 + 2k + 1
⇒ 1 + 3 + 5 + ... to (k + 1) terms = (k + 1)2
Thus the statement is true for n = k + 1 under the assumption that statement is true for n = k
Therefore, the statement 1 + 3 + 5 + ... to n terms = n2 for every positive integer n.