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Question

P(n): 1 + 3 + 5 + 7 + 9 + .... + (2n - 1) = n2 is true for which of the following statements?

A
n N
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B
n N-{1}
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C
n Even numbers
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D
n N - {4}
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Solution

The correct option is A n N
Consider the statement P(n).P(n): 1+3+5+...+(2n1) =n2 When n=1, LHS=1 and RHS=12=1 P(1) is true.
Let the statement be true for some positive integer k,i.e., 1+3+5+...+(2k1)=k2 ...(1)Then we have to prove that P(k+1) is true.Consider, ( 1+3+5+...+(2k1))+(2k+1) =k2+2k+1 (Using (1)] =(k+1)2Thus, P(k+1) is true whenever P(k) is true.Hence, by principal of mathematical induction,P(n) is true for all nN.




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