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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 false

Solution

Both statements are false because Statement 1 does not hold for all natural numbers n, as the left side grows quadratically while the right side grows faster than quadratically for large n. Additionally, Statement 2 is false since n(n + 1) is always divisible by 2, but not necessarily by 3, depending on the value of n.

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