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

(a) S(1) is correct 

(b) S(k) ⇒ S (k + 1) 

(c) S(k) \(\neq\) S(k + 1) 

(d) Principle of mathematical induction can be used to prove the above formula.

1 Answer

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Best answer

Answer: (B) S(k) ⇒ S(k+1)

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

Putting k = 1 on both the sides, we get 

LHS = 1, RHS = 3 + 1 = 4 

LHS ≠ RHS 

⇒ S(1) is not true. 

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

Then, 

To find S(k + 1), add the (k + 1)th term = (2 (k + 1) – 1) = 2k +1 on both the sides of S(k). 

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

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

⇒ S(k + 1) is also true. 

∴ S(k) ⇒ S(k + 1) is true

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