Let S(k) = 1 + 3 + 5 + .. + (2k – 1) = 3 + k^2 . Then which of the following is true

• Mathematical Induction

Let S(k) = 1 + 3 + 5 + .. + (2k – 1) = 3 + k² . Then which of the following is true?

1. principle of mathematical induction can be used to prove the formula
2. S(k) implies S(k + 1)
3. S(k) implies S(k - 1)
4. S(1) is correct

Answer

S(k) = 1 + 3 + .. + (2k – 1) = 3 + k²

When k = 1, L.H.S of S(k) = 1 and  R.H.S of S(k) = 4.

So, S(1) is not true.

Now, S(k + 1); 1 + 3 + 5 + .. + (2k – 1) + (2k +1) = 3 + (k + 1)²

Let S(k) is true, 1 + 3 + 5 + .. + (2k – 1) = k² + 3

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

The correct option is B.