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Let S(k) denote the sum 1 + 3 + 5 +... + (2k - 1), and suppose it is given by 3 + k². Which statement below is true?

1. The formula can be established using the principle of mathematical induction.
2. S(k) implies S(k + 1).
3. S(k) is equivalent to S(k + 1).
4. S(1) is true.

Correct answer: S(k) implies S(k + 1).

Solution

If S(k): 1+3+...+(2k-1)=k^2+3 is assumed, adding (2k+1) gives k^2+3+2k+1=(k+1)^2+3=S(k+1), so S(k) implies S(k+1). However S(1) gives 1 = 4 which is false, so the inductive step holds but the statement is never actually true.

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