Question

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

विकल्प

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

• S(K) ⇒ S(K + 1)

• S(K) `\cancel⇒` S(K + 1)

• S(1) is correct

Answer 1

S(K) ⇒ S(K + 1)

Explanation:

S(K) = 1 + 3 + 5 ... + (2K – 1) = 3 + K²

S(1) : 1 = 3 + 1, which is not true

∵ S(1) is not true.

∴ P.M.I cannot be applied

Let S(K) is true, i.e.

S(K) : 1 + 3 + 5 .... + (2K – 1) = 3 + K²

Adding 2K + 1 on both sides

⇒ 1 + 3 + 5 .... + (2K – 1) + 2K + 1

= 3 + K² + 2K + 1

= 3 + (K + 1)²

= S(K + 1)

∴ S(K) ⇒ S(K + 1)