Principle Of Mathematical Induction Question 47

Let $ S(k)=1+3+5…+(2k-1)=k^{2}. $ Then which of the following is true?

Options:

A) Principle of mathematical induction can be used to prove the formula

B) $ S(k)\Rightarrow S(k+1) $

C) $ S(k)\not{\Rightarrow }S(k+1) $

D) $ S(1) $ is correct

Show Answer

Answer:

Correct Answer: B

Solution:

  • [b] $ S(k)=1+3+5+…+(2k-1)=k^{2} $ $ S(1):1=1, $ which is true $ \because ,S(1) $ is true $ \therefore $ P.M.I cannot be applied Let S(k) be true, i.e, $ 1+3+5…+(2k-1)=k^{2}+1 $ $ \Rightarrow 1+3+5…+(2k-1)+2k+1 $ $ =k^{2}+2k+1+1=(k+1)^{2}+1 $ $ \therefore S(k)\Rightarrow S(k+1) $



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