Let S(K) = 1 +3+5+..... (2K-1) = 3+K². Then which of the following is true?

• S(1) is correct

• Principle of mathematical induction can be used to prove the formula

• S(K) ≠S(K+1)

• S(K)⇒ S(K+1)

D.

S(K)⇒ S(K+1)

S(K) = 1 + 3 + 5 + ...... + (2K - 1) = 3 + K²
Put K = 1 in both sides
∴ L.H.S = 1 and R.H.S. = 3 + 1 = 4 ⇒ L.H.S. ≠ R.H.S.
Put (K + 1) on both sides in the place of K L.H.S. = 1 + 3 + 5 + .... + (2K - 1) + (2K + 1)
R.H.S. = 3 + (K + 1)2 = 3 + K2 + 2K + 1
Let L.H.S. = R.H.S.
1 + 3 + 5 + ....... + (2K - 1) + (2K + 1) = 3 + K² + 2K + 1
⇒ 1 + 3 + 5 + ........ + (2K - 1) = 3 + K² If S(K) is true, then S(K + 1) is also true. Hence, S(K) ⇒ S(K + 1)