Let S(K)=1+3+5+...+(2K-1)=3+K^2. Then which of the following is true? - Mathematics MCQ AIEEE 2004 2004 | ExamTest.live

hardMCQAIEEE 20042004Mathematics

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

Solution & Step-by-step Explanation

The statement given is S (K) = 1 + 3 + 5 + ... + (2 K - 1) = 3 + K^{2} .Check S (1) : LHS = 1; RHS = 3 + 1^{2} = 4. Since 1 \neq = 4, S (1) is incorrect.Assume S (K) is true: 1 + 3 + 5 + ... + (2 K - 1) = 3 + K^{2} .Then for S (K + 1) : L H S = [1 + 3 + 5 + ... + (2 K - 1)] + (2 (K + 1) - 1) . = (3 + K^{2}) + (2 K + 1) . = K^{2} + 2 K + 4 = (K^{2} + 2 K + 1) + 3 = (K + 1)^{2} + 3 . This matches the form of the RHS for K + 1. Thus, although the formula S (K) itself is false for any K, the implication S (K) \Rightarrow S (K + 1) is logically true.

Try it yourself before checking the explanation above.

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

S (1) is correct

Principle of mathematical induction can be used to prove the formula

S (K) \Rightarrow S (K + 1)

S (K) \Rightarrow S (K + 1) is true, although S (K) is false

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