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Consider the following statements for n ∈ N: Statement 1: 1 + 2 + 3 +... + n < (1/8)(2n + 1)². Statement 2: n(n + 1)(n + 5) is divisible by 3. Which of the following is correct?

1. Only Statement 1 is true
2. Only Statement 2 is true
3. Both statements are true
4. Both statements are false

Correct answer: Both statements are true

Solution

Statement 1: n(n+1)/2 = (4n^2+4n)/8 < (4n^2+4n+1)/8 = (2n+1)^2/8, true for all n. Statement 2: n, n+1, n+5 cover all residues mod 3 (since n+5 ≡ n+2), so the product is divisible by 3. Both are true.

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