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Let T(k) denote the claim that 1 + 3 + 5 +... + (2k - 1) = k² + 10. Which statement below is correct?

1. T(1) holds
2. If T(k) is true, then T(k + 1) is also true
3. T(n) is true for every n ∈ N
4. All of the above are correct

Correct answer: If T(k) is true, then T(k + 1) is also true

Solution

The true sum is 1+3+...+(2k-1) = k^2, so T(k): k^2 = k^2 + 10 is never true (T(1): 1 = 11 is false). However, assuming T(k), adding (2k+1) gives sum = k^2+10+(2k+1) = (k+1)^2+10 = T(k+1), so the implication 'T(k) => T(k+1)' is correct even though no T(n) actually holds.

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