Consider the following statements: 1. 2+4+6+...+2n=n^{2}+n 2. The expression n^{2}+n+41 <strong>ALWAYS</strong> gives a prime number for every natural number n Which of the above statements is/are correct?

Explanation

For statement 1, the sum of the first n even natural numbers is an AP with sum \frac{n}{2}(2 + 2n) = n^2 + n, which is true. For statement 2, substituting n = 41 into n^2 + n + 41 gives 41^2 + 41 + 41 = 41(41 + 1 + 1) = 41 \times 43, which is a composite number. Thus, Statement 2 is false.

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